Does turning off your screen while playing Pokemon Go with mobile data on take up ANY mobile data whatsoever?

I have a 250mb plan and would like to save as much data as possible. I have the samsung a32 5g.

I sometimes use it while on public transit and would like to know if turning off the screen while the app is running in the foreground will not use any data whatsoever since I don't get a vibration for nearby Pokemon.

Does anyone know if the battery saver screen also saves data or just power?

I believe that keeping the app opened with the screen turned off still uses some data, for example your GPS position is always tracked for egg-hatching, and you can still receive notifications for raid invitations, defeated gyms or whatever.

Since you have only a 250MB plan I did some research:

According to Business insider, on average Pokémon Go eats up about 25MB in an eight-hour period. This works out at about 3MB per hour.

This Reddit user says over 7MB per hour for a very active level 40 player.

Another Reddit user plays PoGo for at least 4-6 hours a day and their total for the last month on both data and wifi for Pokemon Go is less than 600MB. (This makes 4MB per hour, more or less.)

Morever, reloading PoGo seems to use a lot of data, since it has to redownload the local game data - spawns, gyms, map, etc...

It really depends on how much you play on daily basis, so I don't think that your first problem is data spent with turned off screen.

Does turning off your screen while mobile data is on and the game on take up ANY mobile data whatsoever?

Yes.

Does anyone know if the battery saver screen also saves data or just power?

The battery saver mode simply dims the phone's screen, but keeps everything running in the background. As such, data usage will remain the same.