The first consequence is the full time noise and lower fuel economy. Other than that, no ill effects.
If you are talking about the stock fan clutch, I have been thinking about drilling a small hole thru it and locking it down with a bolt. My clutch don't work either.
However, haven't done that yet. I have a 16" electric fan that will be going on very soon. What I like about this idea is that when the wx is very hot, I can have it run full time for good a/c cooling at idle. And, can turn it off when not needed.
Lock up the fan clutch, and you have no options left.