Normally it does not affect stability of protein as both of them are different folding units (foldons). However, for safer side one can have linker.

It's hard to predict whether the stability is affected by the fusion however its good to keep in mind that linker regions often are more prone to protease activity. This could result in a heterogenous sample of tagged and untagged proteins. 

The in vitro stability may not be affected depending upon the protein or active moiety fused to GFP but accelerated stability studies would need to be done and stability indicating forms determined (such as aggregates).  The formulation may also require some development.  The in vivo DM will be dependent upon the structural features of the active moiety and the PK and the target site(s).  Some investigators have used incubation with serum to look for degradation but this is unreliable since it does not necessarily reflect what happens at the target site.  Will you be able to firstly determine if the GFP causes any steric hindrances or activity changes using in-vitro assays? Linker size, stability, and sequence are relevant to this and this is a field of study on its own.  Before PK is determined there will need to be exposure studies.  However, the molecular or hydrodynamic radius around 70 kD is good for avoiding clearance by glomerular filtration.  So it's the combination of both moieties that influence PK.  Also the amount of acidic residues influence the filtration rate.  If the protein species is heterologous to the test animal then there's a risk of immunogenicity clearance if the same animal is used multiple times.  You might opened a whole lot of work for yourself.