Just as an aside, I'm guessing that some of the 'grease' you're drinking might really be gelatin. If I roast a joint with any bone or connective tissue in, then once the run-off oily juices have cooled they tend to solidify into a big block of gelatin (great for stock or gravy) and a layer of solidified fat that's risen to the top.

There's nothing wrong with large amounts of animal fat in itself- it should probably form the majority of your calorie intake. Consuming more than you need is still going to be negative though. You also might want to watch the omega 6: the beef will be pretty great, the pork middling but the chicken might be rather heavy on omega-6 even if they were raised well. You might find it slightly easier to eat more calories than you need if you're just chugging down a liquid, in the same way that eating lots of cream will be a lot easier than devouring butter or lard, but this is unlikely to be a massive practical problem.

I like Robb Wolf's assessment of things:

How do you look, feel and perform?

If you change your diet, and you look, feel and perform better. Then it was likely a good change. If you're nervous about the amount you consume, try increasing your consumption or decreasing your consumption and see if you look, feel and perform, better or worse. It's important to pay attention to your body, and take pictures! ;)

Well, wouldn't part of that be drippings or juice instead of straight fat?

It might help to know if you are normal weight or if you have a weight problem. I eat the stuff too, but usually use it in a sauce to pour over the meat, and others help me eat it.

If you are healthy and still able to outrun a wooly mammoth, I don't see a problem with your unusual habit. :)

It ain't grease. It's saturated fat. And it's good for you.

The raw paleo people eat all their meat without cooking and thus they eat what you have forced out of the meat by cooking.

I have tried raw meat and it is ok, but I really like the taste of hot rare meat and fat better than cold and raw.

I would drink that liquid fat with relish. It also makes a good salad dressing.

And I prefer Michaal Ruhlman's definition of grease from The Elements of Cooking:

Among the most nefarious by-products of cooking, grease refers to the airborne particles of fat that rise from hot cooking pans and collect on whatever they can find. Grease should not be used snynonymously with fat. Fat is good; grease is evil.

If you are in good shape and have no other problems, I say go for it!

As others have stated the "grease" may contain more than just fat. You can figure this out by pouring it into a jar and putting it in the refrigerator. The fat should separate from the watery (and hopefully gelatinous) components. You have the option of reducing the watery component into a great sauce.

I don't know of a limit on gelatin and the watery component of "grease". As for fat, you will know when you consume too much. Your body can only digest so much at once- the rest will pass through.

I would only consider your habit unusual because you are consuming it separately after the meal. Normally that would be used as something like gravy to dip the meat in. It is a shame to let that part go to waste, and a hundred years ago I doubt anyone would have.