I know this question will vary depending on location. But if I were to put up a 300lb gravity fed protein feeder in Coleman tx, about how long should I expect it to last before it is empty.

Once they know it's there and are use to it, not long. Between January and August probably no more than 2 weeks depending on deer density, how many others are feeding protein and other things. Put it near a water source.

one week for me - 8 deer

I'm not sure how good that will be for me since I can only make it to my lease a month. Would it even be worth it to put out?

If your goin once a month I would say you need at least 600lbs.

It will depend on how many deer you are seeing at that location. If you are seeing 20 deer then it will last 6-7 days once they get on the feed good. Less deer it will last longer and more deer will empty it much faster. At first it will last longer than when they have been on feed a while and it gets really dry. 2 yrs ago I had one 275# cap feeder going empty every 6-7 days in the summer months. Last year it lasted a month. Same feeder and same amount of deer just it rained 45" and I had good food plots.

A timed protein feeder might be a good choice if your time at the lease is limited. Not as good as free choice, but at least you'll know how long you'll be giving protein.

I recommend at least a 600lb feeder, really a 1,000lb feeder is ideal.