Atmospheric Nitrogen gas and human body pH buffers
One chemical equation that can be written for nitrogenase is this one:

$\ce{N_2 + 8 \ H^+ + 8 \ e^- + 16 \ ATP \longrightarrow 2 \ NH_3 + H_2 + 16 \ ADP + 16 P_{i}}$

The equation shows that not only does nitrogenase generate the basic molecule $\ce{NH3}$, it also consumes protons (i.e. acid) to form hydrogen as an inescapable byproduct of its mechanism.  

So yes, in theory nitrogenase could perturb the human body's ability to buffer pH.

However, the relative rates are always important.  Human requirements for nitrogen are on the order of 80 grams of protein per day.  This translates to about 12 g of (fixed) nitrogen atoms per day, which corresponds to 0.85 mol of N.  If all of that nitrogen were to come from nitrogenase, then 0.85 &times; 8 &div; 2 = 3.4 mol of acidity would be lost via proton consumption in addition to the 0.85 mol of base generated as $\ce{NH3}$, or about 5.2 mol of "base" in total.

But human pH homeostasis works even though humans breathe out about 900 g of an acid equivalent ($\ce{CO2}$) per day, or about 20 mol.

So even if nitrogenase completely replaced human nitrogen requirements in some hypothetic bioengineered human, the demands on pH homeostasis would only change by about 25%, maximum.  And considering that pH homeostasis still works even when humans are exercising vigorously, when "instantaneous" $\ce{CO2}$ production rates are far higher, I doubt even a 25% change would be too much to handle.

Thus in conclusion I don't think nitrogenase would wreck human pH homeostasis, unless its activity in the human body was increased to a level far beyond what is necessary to replace dietary N requirements.