As a big fan of this genre I want to like this game, but the input lag I'm experiencing is really hurting the feel of the game and making it difficult to handle the ship. Does anyone know if there's a way to fix this?

Try changing your DOSBox settings, particularly the number of cycles. Beside this, I think you can try different graphics mode using the setup utility.
Take a look here for suggested settings.

You may want to try OpenTyrian if you can't get Tyrian 2000 set up. OpenTyrian is a source port of the original Tyrian and uses SDL as input handling. It's a solid port in my experience although I haven't tried it on Windows yet.

dr.schliemann, thank you. This is good advice for DOSBox games in general and usually sorts various problems out.
Unfortunately (after trying various setups), though I've improved the overall look of the game, I am still having no luck reducing it's laggy feel, sadly.
The game feels very "slippery". Playing using either keyboard or mouse (acceleration turned off).

Messing with some other settings did fix another problem; audio lag... that's obviously separate.

zaine-h, thank you too. Never heard of that before, so I shall have a look at it.

I've tried everything I know and it still feels 'off'. The funny thing is, I'm playing on 'Normal' game speed and noticed that by switching to Turbo-Mode, there is barely any noticeable delay. Meaning it may just be related to the in-game speed setting.

Maybe it's just the case that this is how the game is meant to be; the ship has heavy inertia and takes a while to come to a stop after releasing a direction key. Still not 100% sure though. If this is how it's meant to be, I'm not too keen on the feel.