I would like to regress basal metabolic rate (BMR) on height, weight, age and gender. How do I do this taking into consideration the interaction between the variable? Wouldn't I have to consider interactions upto fouth order?
Theoretically yes, you could go up to fourth-order interaction, but practically we rarely go beyond second- (or perhaps third-) order interactions, I think. They are very hard to assign any meaningful interpretation (i.e. medical meaning).
To test whether interactions are needed, you can simply use a Wald-type test or LM-test, as models with/without interaction represent nested models.
(For large samples they might be always significant, in such scenarios it might also be useful to have a look at how $\chi^2$ changes when the interaction is added.)
And of course, it is always important to consider -- in addition to the statistical points -- how meaningful these interactions are from the clinical point of view.