What are the current practices for sampling of biological tissue and how can medical staff appropriately prepare their working environment to reduce contamination of samples?

At the sampling level, use sterile instruments and containers and avoid all potential sources of cross-contamination; minimize handling time and try to keep samples at 4ºC (or lower), for sample integrity. Having no experience in the medical field, but assuming human tissue samples, chain of custody and traceability are likely also key aspects to consider. 
At the analytical level, consider the already noted issues: contamination prevention, use of non-plastic material, whenever possible, use of safety cabinets. Procedural blanks and recovery controls are also necessary. 

Sampling of biological tissues is outside my expertise but I would recommend ensuring medical staff involved in taking samples are fully appraised of the basic research goals of the study to maximise their "buy-in". If at all possible, plastic should be avoided in sample collection/storage. If this is not possible - e.g. standard kits for blood sample collection involve plastic, then the extent of sample contamination should be assessed. For example, pig blood should be drawn through the blood sample collection kit used for human sample collection, analysed for MNPs and the results compared to that obtained for the same sample without contact with the sample collection kit.

The current practices for sampling of biological tissue  are  composed first of  the pre-sampling protocol which involves the informed consent that ensures the proper ethical approvals in the case of animal studies and patient consent in the case of human studies, the correct data labeling of the containers, the preparation of the instruments that are to be used, the preparation and use of proper personal protective equipment, and the set-up of an aseptic field. Secondly, there is the case of the sampling methods which requires the use of a sterile technique, the use of appropriate tools in order to avoid thermal or mechanical degradation of the samples, immediate storage (the fresh tissues should be freeze using liquid nitrogen or stored on dry ice depending on the intended applications, histological samples needs to be fixed using proper fixatives, again depending on the intended applications, samples for molecular analyses should be stored in stabilization solutions),  and proper documentation (the samples need to be properly labeled, a log of chain of custody and sample condition such as storage time, transport details, or any change of the standard protocols should be noted). 
In order to try and reduce contamination of samples the medical staff should assure a sterile environment (this can be achieved by disinfecting the surfaces with appropriate biocidal cleaning agents after each sampling and also by using  a clean room for processing), to maintain a proper control of the airflow (this might be assured by reducing the  personnel to reduce airborne contamination, and to minimize the airflow over the samples), to assure an adequate personal protective equipment (this involves changing gloves between samples, using hair nets and face masks and avoiding touching non-sterile surfaces with the gloved hands). Also, another important measures that the medical staff should consider are the ones regarding the instrument management (try and use separate sterile instruments  for each sample and decontaminate between uses when appropriate and dispose of  the ones that are intended for single use) and sample transport (for this a clear separation of samples and appropriate conditions  during transport is necessary as well as the use of sterile and leak proof containers). Additionally  the regular training and inspections are required ( organize sessions of staff training in biosafety and decontamination techniques and substances and also  try and do as often as possible quality control tests using blanks or controls).

The key issue is that medical staff is not always trained to apply methods and approaches that reduce contamination of samples, let alone that they have the mindset to work in a plastic-free environment. Most likely, this is something that cannot be expected from medical staff and actually most medical staff is focussed on maximizing plastic exposure, for instance in their training of using gloves and replacing them whenever needed.
Part of the way out, is to ask medical staff to take as much as a sample as possible and to make sure that the sample is left intact as much as possible. On top of this, plastic-free equipment is to be used befofe, during, and after sampling and storage of the tissue in the hospital. In the ideal situation, medical staff would prepare their working environment as is commonly done in industrial clean rooms.

This is a broad question as the term 'biological tissue' is very diverse and can range from plankton to human specimen to whale liver.  Most of the studies of MNPs using biological specimens have relied on samples collected by a collaborator, who, sometimes are not experts in environmental analysis.  Sometimes archived samples from the past studies over the years have been used.  In that case storage conditions and contamination during storage are not known.  This is a serious concern.  This is not to demean the collaborators in other areas of expertise, but to warn MNP investigators that sample collection is a critical step in environmental analysis.  A compromised sample will result in irreproducible data, irrespective of how good your analytical method is.  The investigator should take responsibility for developing appropriate protocol for collaborators and educate them about sources of MNP contamination and how to avoid contaminating samples at the time of collection.

Term 'biological tissue' is very diverse and can range from plankton to human specimen to whale liver.  Most of the studies of MNPs using biological specimens have relied on samples collected by a collaborator, who, sometimes are not experts in environmental analysis.  Sometimes archived samples from the past studies over the years have been used.  In that case storage conditions and contamination during storage are not known.  This is a serious concern.  This is not to demean the collaborators in other areas of expertise, but to warn MNP investigators that sample collection is a critical step in environmental analysis.  A compromised sample will result in irreproducible data, irrespective of how good your analytical method is.  The investigator should take responsibility for developing appropriate protocol for collaborators and educate them about sources of MNP contamination and how to avoid contaminating samples at the time of collection.

Modern medicine practice is saturated with plastic PPE, disposables, and packaging. A through checklist of the materials in the sampling arena, sample containers/packaging, transport/receiving, etc. will guide the analyst in understanding possible sources of contamination and specific MNPs attend to in the sample, as well as the field and process blanks.
"That which is measured improves, that which is measured and reported improves faster" (paraphrasing Thomas S. Monson, Karl Pearson)

At the sake of being redundant... training, training, training (even if minimal)

I am not expert in medical space. From reading literature, biological tissue is either from stored medium in early sampling or following medical procedures, which mainly considering safety and meeting medical treatment goals. In my opinion, medical staff should still consider medical purpose and procedures as priority. When circumstances allow and options are available , using alternative non-plastics tools and consumables instead. Microplastics in the surgical environment - ScienceDirect 

For human tissues, medical facilities are filled with plastic materials. If samples are to be taken for microplastic measurements, much greater controls are needed before, during, and after the medical procedure.  The same best practices as listed in the first section should be used.
If the review is meant to focus on human tissues, that should be explained and wording on environmental samples can be diminished.

1. When sampling biological tissue, non-plastic tools, glass or stainless steel, should be used to avoid introducing polymers. 
When sampling biological tissue, the sampling environment should be as clean as possible. 
Biological tissues should be stored at low temperatures. 

2. Medical staff should ware non-polymer lab coats and nitrile gloves. The working surfaces should be cleaned with proper cleaning reagent. Glass pipettes and containers should replace plastic ones. Filter all liquids with glass fiber filters. Include different types of blanks in the experiments. label the samples well to reduce handling errors.