What are the major laboratory sources for background MNP contamination and what standard practices need to be implemented in laboratories to keep the background contamination to minimum?

Laboratory reagents, equipment, PPE, and air. Standard practices to minimise contamination should include avoidance (or minimisation when not possible) of plastic equipment (use glass or metal instead). Also consider conducting work in a clean air hood with ideally (thought noting cost implications) working in a clean room with filtered air. Use of cotton lab coats should be obligatory, as is regular, frequent (e.g. daily) washing of lab surfaces with ethanol. 

The major sources of contamination are fibers, usually from clothing, but also from air conditioning units. Other sources include plastic labware, contaminated reagents,  packaging materials. Minimal sources could include cross contamination from prior analyses. Minimize the use of plasticware, use of natural fibers (e.g., cotton), use of safety cabinets, filter reagents, clean surfaces and material rinsing are all recommended practices. 

 The MNP contamination has multiple sources contributing to background interference in major sensitive analyses. The major sources are coming from airborne fibers that could come from synthetic lab coats and also from dust from plastic equipment, shedding from plastic labware and consumables, contaminated water systems such as ultrapure or Milli-Q sources, and contact with plastic-based surfaces and instruments. More than that, the  human contamination  that could arise from clothing, hair, and improper handling also poses a risk. In order to be able to minimize these contaminants,  the laboratories should implement standard practices such as working in clean air environments, using glass or metal equipment where possible, and filtering all reagents and water. Complementary measures could include the incorporation of procedural blanks, switch to the use of cotton-based PPE,  the regularly cleaning  of the surfaces with filtered solvents, monitoring  the environmental contamination, and minimizing sample exposure to open air. 

In principal, anything that is present in the laboratory can serve as a source for background MNP contamination unless it has been verified that this is not the case. This ranges from solvents, any lab equipment, ceiling materials, staff working in the laboratory, etc.
The key issue is that laboratory staff and management need to have a constant focus on possible sources of contamination and on minimizing MNP contamination. This is comparable to working in a clean room, and it is the mindset that is in itself the key issue.
A required standard practise is, apart from technical efforts like including blanks, to maintain the mindset and take the appropriate precautions. One of the issues is in this respect the need to critical inventorize possible sources and actively secure the mindset.

Ambient air is a major source of contamination. It contains MNP from all the plastic sources inside and nearby outside of the laboratory. As stated, laminar flow hoods help. 
Samples can be contaminated even if they are "covered". Covering that are not air tight will allow the sample containers to "breathe" as the laboratory temperature cycles during the day/night. This will bring MNP into the stored samples.  Recommended: airtight storage containers kept in refrigerator or freezer which limits magnitude of temperature cycles.

Major laboratory sources for background MNP contaminations are containers, tools, instruments, and PPEs made of synthetic polymers. Standard practices are to use alternative materials such as metal and glasses and use dedicated area and setups and MNP study.

Dust, in general, is the biggest source and most of the microplastics in dust of a typical lab will come from clothing. 
opening samples only inside hepa filtered air
nonplastic clothing or labcoats
minimum or no use of gloves
filtering all reagents
cleaning glassware and metal instruments thoroughly - rinsing with <1um filtered water and baking them at 400 oC or higher
keeping clean glassware covered with baked foil
general lab cleanliness (wet dusting and vacuuming floors when samples are NOT being processed)

Major laboratory sources for background MNP contamination are air, reagents, equipment, glassware, containers, vial caps, gloves, lab coats, filters.
Select stearate free gloves; cover the samples; use solvent cleaned glassware or metal; be cautious of the cap materials as most caps contain plastics and should chose those without targeted analytes.

Although not discussed in the manuscript, one major source of contamination is sampling devices and contamination while sampling biospecimens.  Most studies that analyzed human specimens such as blood, brain, and placenta received samples from physicians or collaborators who collected samples.  The MNP levels reported in those studies were <10 particles per tissue or per unit mass.  However, such studies can be sensitive and often receive media.  Data quality of those studies needs to be strictly validated.  Those samples were collected in an environment that could easily contaminate blood or placenta tissues (in hospital settings where physicians use plastic products during surgery).  It is important that study investigators who are familiar with such contamination issues should be involved in sample collection.  

Establishing a lab environment that is free of MNPs is very important.  A difficult source of avoid MNP contamination is airborne particles.  Controlling/monitoring lab air quality is critical and ISO clean 5 or class 6 clean rooms may help reduce contamination by airborne particles, but such infrastructure can be expensive.  This is critical for labs measuring MNPs at trace levels, especially in human specimens.  A clean room, free of plastics, should be used in sample preparation.  Products including collection and storage devices, sample extraction tools, that come into contact with samples should be free of plastics.