What are the advantages/disadvantages of the different regression approaches when using epidemiology exposure-response data to estimate cancer cancer potency?

Tauqeer Hussain Mallhi; The type of regression used depends on the outcome variable. Different independent can be used. They can be continuous, dichotomic or categorical (normally in a categorical variable we define one reference) .
When analyzing continuous variables as independent we consider that each algorism will modify that OR. As an example, in one paper published in 2017, we analyzed factors that interfere with surgical site infection. Analyzing surgery length, in hours, we found that each minute could contribute to the rise in the incidence of Surgical Site Infection. Each hour presented an OR of 1.3, or 30% per hour. 
There is no trouble to do that in statistical programs such as Stata, SPSS and R. Just use the code for logistic regression and put your continuous variable.

Thanks for the information. Can you please guide me that whether we can use logistic regression when dependent variable is bivariate and independent is continuous?

There are many things to consider in terms of advantages/disadvantages of different regression models - making this a huge question to answer.  However, one of the first things to consider is properties of the response (or dependent or outcome) variable.  Different types of regression models (such as those listed) are based on different probability models that should align with properties of the response variable.  For example, Poisson log-linear regression is a model for responses that are integer counts per some unit effort, while Cox proportional hazard is used to model responses that are survival times.   Before making any choice, it's important to understand the assumptions underlying each model and assess their reasonableness given the problem and data in question.  We tend to assume models are free, but model assumptions insert information into an analysis beyond that contained in the data - and are best justified when consistent the design, the data collected, and the objectives of the research.