Who was on the phone in Memento?

While I haven't seen it for quite some time, I'm pretty confident to say, that it isn't ever revealed in the movie and the audience is just not supposed to know.

The reason why Leonard has that tattoo saying "never answer the phone" is, that his desease doesn't work that well over the phone, since after some time he doesn't really know who he's talking to, which is shown in one scene when Leonard finally asks the phone "wait, who are you?" (or something similar). He just doesn't know and probably hasn't known for the majority of the whole conversation. And in the same way the audience doesn't know who he's been talking to the whole time. This fits to the rest of the unconventional narrative structure of the movie, which more or less repeatedly puts the audience into the same situation as Leonard, not knowing what just happened some minutes ago.

So by keeping the audience unaware who he's actually talking to all the time, the movie stays in line with its whole motive of not giving the audience too much more information than the anterograde amnesian protagonist. And not even revealing who was at the phone at the end also fits to the disillusioning ending (beginning?) of the whole story, I think, which doesn't present Leonard with the murderer he (and the audience) searched all the time, but just with the very plain and sad truth of yet another unimportant victim in Leonard's struggle for mental occupation. The man on the phone is as unimportant in understanding the real case of his wife's death as all the other clues acquired by Leonard throughout his "pursuit".

EDIT: After rewatching it, I would ammend this rather common sense answer by some substance based on the movie's actual story. It was indeed most of the time Teddy who he's talking to. This is not only evident from him repeatedly calling the other person "officer" (which we learn at the end/beginning Teddy is supposed to be).

There is a long phone call when Leonard is starting to talk about Sammy Jenkins and this is also the talk when he learns about John G.'s background as drug dealer, information that obviously comes from Teddy, who we learn later wanted to get Leonard on the track of Jimmy the drug dealer. This call lasts until Leonard discovers the "Never answer the phone"-tattoo and hangs on, hesitating to pick up the phone for some time. Then he gets a message under the door saying "answer my calls", accompanied by the Polaroid photo showing the half-naked Leonard after his first kill (as we learn at the end/beginning), which we know also has to come from Teddy. And indeed this last phone call ends with Leonard saying "You're in the lobby? I'll be right there", which is followed by Leonard meeting Teddy in the hotel lobby.

But most of my above points about the meaning of the unknown phone caller still stand, I think, since it is only at the very end of the movie that we get all the information to know that it was Teddy he was talking to most of the time.

I am afraid that I can't add more than what is already explained in IMDB FAQ:

In some cases it is clear that Leonard is talking to the front desk (and at least one time it is Burt since Leonard says "Burt" as if the man on the phone indicated his name). The other times it is most likely Teddy; early in the film Leonard tells Burt to hold all his calls except Teddy's as an exception. Thus it appears to be Teddy who is telling Leonard that Jimmy Grantz is the 2nd attacker on the phone. Teddy does admit to doing this at the end of the film as well, supporting the conjecture that Teddy is on the phone.

If you have any question regarding this explanation, you can comment on the answer.

The person who he was talking to on the phone was Teddy. Leonard was talking to an officer which Teddy was an officer, explaining the situation regarding Sammie. Teddy as you know has heard the stories a thousand time (which he told Leonard at the end.) While they were having the conversation, Leonard was flipping the stack of documents he has relating to the murder of his wife (which Teddy gave it to him, he also explained in the end.) When he saw the tattoo, he freaked out and hung out and refused to pick up the phone, but a photo of Leonard was slide under the door (which Teddy also mentioned to Leonard at the end that he took of him when Leonard killed the "murder" of his wife.) And the officer on the phone was also giving information to Leonard referring the killer was a drug dealer that lead him to Jimmy's warehouse (which Teddy wanted him to kill Jimmy.) At the end the officer asked him to meet him outside in the lobby, and Teddy appeared. Then Leonard took a picture of him and was about to write "officer name", but Teddy told Leonard to just write Teddy. So, it was Teddy that he was talking on the phone.