I see these two words misused constantly.
“Premiere” is a noun. Webster’s defines it as “a first performance or exhibition.” Premiere can also be a verb, but more on that in a sec.
“Premier” is an adjective! Webster’s: “first in position, rank or importance.”
Thus, you can have the premier conductor of Mahler giving the premiere of a newly discovered work by the composer.
Premiere is often, and apparently correctly, used as a verb, as in “the symphony orchestra premiered a new work by Michael Daugherty.” But premiere wasn’t always accepted as a verb. I had a professor in college, none other than Piero Weiss, an astounding language expert, who would not allow the use of premiere as a verb, and in fact practically blew a gasket when I did. To this day, I do not like to use premiere as a verb, and instead would write the sentence above as “the symphony orchestra gave the premiere of a new work by Michael Daugherty.” Then I’d say how much I hated it.
By the way, I also think that ensembles that play symphonies are best referred to as orchestras, or symphony orchestras, rather than symphonies. It’s more elegant, and it avoids confusing the reader. Philharmonic is also an adjective, but a lost cause. I refer to the Los Angeles Philharmonic or the Berlin Philharmonic with impunity, though technically both those groups should properly have the word “Orchestra” after Philharmonic. The Pacific Symphony Orchestra officially changed its name to the Pacific Symphony a few years ago, but wouldn’t have were I in on the meeting. The Pacific Symphony sounds to me like a piece that Michael Daugherty might write.
Note: Yes, premier can be a noun when you are referring to a prime minister, as in Premier Khrushchev.