I have a 340 Cuda I want to pull the engine on. Question is it has TTI headers on it, do I have to remove them to get it out or can they just be moved aside?

I've done it that way. Just see if you could spread them. The problem is the motor might turn as it leaves the mounts and doing damage. You'll probably need to take the starter out first doing it that way.

If I unbolt the headers at the collectors will they come out with the engine?

they'll want to take out firewall and everything with them, I'd unbolt and collectors, and at the heads, wrap a couple old towels around them and when you begin pulling motor, just take your time and make sure each side shakes loose.

Once you clear them off the motor...pull 'er out.

The headers in our cars are designed to fit around the unfriendly obstacles like torsion bars, steering boxes, a starter and oil filters. The only way to pull the motor with the headers still on is to lower the motor and K member from underneath the car.
That said, if you want to keep the headers in the car while pulling the motor, it can be difficult if you are trying to keep everything scratch free and looking nice. The 4 outer exhaust manifold/header mount studs may make it difficult. As stated before, you can try to spread the headers apart from the motor, but to do that you will also need to at least loosen the collector bolts to allow some articulation. Its a great idea to wrap the headers with towels or thick garbage bags to avoid scratches. Keep us informed. I hope this helps.