It's difficult to say.
To really understand, you have to look at the whole movie including the ending (which I will spoil in this answer).
A full (also spoilerific) breakdown of the movie's plot can be found here, but ultimately we see throughout the film that he has these telekinetic, supernatural moments - but all are revealed to be hallucinations, time and time again.
For example, we see Riggan using telekinesis to throw objects around his dressing room - until his lawyer enters and we see Riggan was using his hands, not his mind.
The film then ends with Riggan starring in the play, getting rave reviews from critics and ending up in hospital after using a real gun in the final part of the film. His injuries leave him needing bandages which cover his face in the same way the Birdman's mask covered his. When he is alone in his room, he spots some birds outside his window. He climbs onto his window ledge and jumps - Sam then rushes in and looks to the ground expecting to see a body, but there isn't one - she looks to the sky and smiles - and that's the end!
So ultimately, this leaves a plethora of possible interpretations of the telekinesis.
It's possible he really is telekinetic and that the film finishes with his flying away in a blaze of glory. However, this is the least likely option as the film continually rebukes this notion, showing these supernatural acts to be false.
In the last stage of the play he actually kills himself with the gun (instead of merely wounding himself). This makes sense, as everything after that is perfect - he gets a rave review for his performance, a standing ovation and a heart-warming moment with Sam. In other words, those are all hallucinations, including the end of the film - he's really dead (or at least on his deathbed).
Everything is real up until the moment he jumps out the window, which is symbolic of him soaring once again and moving on with his life.
Ultimately, we just don't know. I would say the second option is most likely (and is certainly my preferred interpretation of the film). However, as one of the film's writers said:
"We're not going to sit around and explain the ending. I guess my
thing is, if you can silence the voice of mediocrity, then what is
possible? [That] is good enough for me," Dinelaris said. "But we
thought if we answered that question at the end, it would seem very,
very small. Is he famous because he shot himself? That's small. Is he
still miserable? That's small. Everything seemed small."
So, I know you wanted an answer explicitly on the telekinesis, but it's impossible to really talk about it without understanding the ending - which, alas, is open to a lot of interpretation.