Leather jacket got wet a week ago, still damp and heavy, should i throw it into the dryer and check up on it repeatedly?

Never use heat to dry wet leather. Put the jacket on a heavy coat hanger in well ventilated area and be patient. It could take days or weeks to dry completely. If it's too cold to hang it outside, too cold to have a window open, then put a small electric fan in the room to keep the air moving around the jacket. 

Sure, what could go wrong?

No, don't put it in the dryer.  Hang in up - by itself - with nothing close to it - so it can dry thoroughly.  

I wouldn't put a leather jacket in the dryer on any setting. I'd leave it be until it dries naturally.

Only use a dryer if there is a suitable program. Too much heat will ruin it. If there is no specific program try the 'airing' one. Yes check it frequently.

It won't shrink but too much heat will crack the leather.