[Allita102]

If you just got a new bird and is scared of you you shouldn’t put a cover over it because it might feel suffocated but if it is used to you already you can TRY to put a cover over it but if they don’t like it just don’t do it.

 

[Redmadhatter]

Depends on your bird, although tiels are prone to night frights, which can be really bad if they get a bleeding blood feather in the middle of the night and you don't notice. You'll mostly have a dead bird by morning if that happens.
For my bird I found out that he sleeps much better at night with a cage cover. However mke sure whatever cover you use, leave either a night light inside the cover or leave a crack on the cover so it's not completely dark.
I suggest at least cover the cage at night/your bird's sleep time. As this seems to encourage a calm state to most tiels I have seen/known.

 

[adelpks]

It depends on your circumstances and your tiel. I've never heard of them feeling suffocated by covering, I think it's a comfort and protection. My tiel wouldn't settle without the cover because he could do see what was happening. But if he was covered and the cover shifted and flapped he would get scared. My vet advised me to put him in a separate room and shut the door, away from distractions and no cover. But this may not be an option for everyone.