I decided to animate all bodies of water and I am curious what this does to BYOND's lag? Is it animations on the screen or global animations? Are animations that aren't on screen still causing lag? I ask because the world map is 600X600 and it's islands in water.

I tested this while playing and I didn't see any huge spike in lag, but that changes when you got players connecting to the world.
If you're talking frames of a single icon_state I would say it's neglectible, unless you're talking 100+ frames, maybe. Might be wrong.
Tiles are animated on the client. BYOND avoids redrawing the screen when there are no changes, but animated tiles will force a screen redraw whenever there is a frame change.

The effect should be completely negligible.
What kind of animations are you talking about? Icon state animations in the icon editor? Or animate() loops?

The server doesn't know what icon animation is. It just tells clients what icon file and state (and the other appearance variables) to use, but the client figures out the frame on its own and uses that frame for all visually identical icons it can see.

It's not like each turf in the world has an infinite loop cycling through its frames over time. The client can quickly figure out which frame to use at any given timestamp.
Whatever issues you're seeing from players connecting won't have anything to do with the animated turfs, but with what the server is doing. I recommend getting some profiling data on the procs that are running and their average times. That can tell you a great deal.
Sorry it took me so long to get back to this.

I only ask because I remember way back in the day, it felt like any time I hit a spot in a game with a lot of animated tiles, suddenly everything would come to a crawl.

That's why I asked. I am not personally seeing any sort of worrying lag yet.