I have a 40w red tube, which seems to have partially lost its vacuum. It does not want to start on preheat and the ends glow orange but putting up a tesla coil up to it makes it glow a little bit (normal 40w tubes glow almost halfway with my tesla coil but this one only has a patch).
What would happen if I connected it to an electronic ballast and is there any way to rejuvenate it?

I don't think you can do anything with it, unfortunatelly...

You could try a neon sign workshop, if that lamp is worth spending 100-1000 bucks on, to you personally.

They have the appropriate equipment to helium test the seals, to repair or replace electrodes and to re-fill the tube with the correct mixture of gases. However, i can imagine that any work will be done without any guarantee whatsoever because repairing a fluorescent is completely different from repairing a "standard" neon tube, with regards to the specific gas filling.

Edit: i see you live in India. I have no clue if traditional neon sign shops exist over there at all.

The problem with Neon shops is, they use to be equipped just for cold cathodes, not hot ones.
The thing is, once you expose an already activated hot cathode to air, it gets ruined by the oxygen. So there is no way they will work after even successful evacuation and reseal, they would have to be replaced (or at least the emission coating).
And because of these complications, plus the fact the signage business usually does not need the high currents the hot cathodes are needed for, most often they are not equipped to deal with them. The cold cathodes are rather foolproof, most of them you can expose to air as you wish, they just need to be degassed and that is it, no need to mess up with any cathode coating chemicals.

I already have another one of this exact lamp so it isn't really that important to me. All I wanted to check was if there can be anything done about it.
Also, I have never seen a neon workshop (if I had I would have gotten a few), only either maintained neon signs or most commonly abandoned (trying to get those too).

I believe you can rather simply recover a hot cathode by re-heating it under vacuum. That way barium/strontium carbonates and hydroxides formed on the contact with moist air will be again decomposed to the oxides, oxides will react with filament base forming some tiny amount of free Ba/Sr metals and so cathode will be re-activated again.

There were at least two attempts to introduce hot cathodes to the neon trade, in USA and in Germany. Generally, both failed because hot cathodes were too cumbersome for a typical neon shop (true). American method by EGL was the simplest in the execution, electrode end blanks had both cold cathode shells and hot cathode filaments, in the processing the tube was first bombarded with an air fill, using cold cathodes, normal neon shop process, then pumped to high vacuum and hot cathodes were activated by applying some current, limited by a large lightbulb in the circuit. The tube then was filled with argon/mercury in the standard way.
 

Update- too bad the filaments broke
Any hope on shorting it out?

Personally I would just feed it to a 1000 watt ballast at this point. Send it off with a zap and maybe a bang!