I have a CES production function (GDP in the data) in my DSGE model, composed of non-energy and energy inputs. I want to estimate the parameters of those inputs using Bayesian techniques. I only have data on energy in BTU (British Thermal Units), not in monetary value. Is it possible to use GDP (or another aggregate) and energy in BTU to estimate the parameters? Is it feasible even if they don’t have the same units? Of course, I will to transform the data per worker and do the detrending later on.

Which parameters do you mean? The elasticity or the share parameters of the CES?
There have been related discussions like Distribution Parameters in CES

For example, there are two inputs inside the CES, which is the final good. And each input has a cobb douglas production function. I want to estimate the parameters of each cobb douglas function. Now, I have data for the non-energy input in monetary value and for the energy input in BTU (difficult to convert into monetary value). Is it feasible to make this estimation knowing that the two variables have different units of measurement ?

It’s hard to answer in general. But my hunch is that this should be feasible. If you have output in monetary value and one input in monetary value, you can compute the value share of the first input. The value share of the second input should then be the complement.