That is quite a general question, and I can't give you a definite answer if you don't give me some other information. For example, are the SM fermions charged under this interaction? If so, with what charges? What are exactly the gauge representations of the new fermions? How do they appear in the Lagrangian?
You see, there are MANY ways of answering these questions, and each lead to a different model. I could give a good example, for example $U(1)$ being a $B-L$ symmetry that arises naturally in $SO(10)$ or large group GUTs. In this case the extra heavy fermions would be the conjugated right-handed neutrinos, with a heavy Majorana mass. This would lead to an anomaly free theory (as the right-handed neutrinos have the desired charges for that) with a sufficiently heavy right-handed SM singlet to produce a see-saw mechanism to generate low physical masses for the neutrinos.
Other than that, the options are literally infinite, even if the interesting ones are significantly less this means that you can have $U(1)$ symmetries with very different phenomena at low energies.
Also, if you are thinking in supersymmetric model building, some $U(1)$ can be added to enfore an R-symmetry, which has many appealing low energy consequences.
Added in reply to
What about case when SM fermions aren't charged under this new group?
Well in that case I think the only obvious scenario (not unique for sure, but this is the first that comes to mind) is that it might be a dark matter model, as in this case the new matter content does not interact with ordinary matter.
Obviously that this statement depends on other things: how heavy are the new particles? What is the full spectrum of the extra states? Etc I am not an expert in dark matter model building - perhaps you have a few papers and ideas in the back of your mind when you asked the question - so it's hard for me to say what are the exact benefits of this type of models over, say, supersymmetric dark matter candidates.
This post imported from StackExchange Physics at 2015-10-21 15:10 (UTC), posted by SE-user romanovzky