The Doctor's exact sexuality is not known, however, it's often implied that he's not straight.
On screen, the Doctor has expressed romantic interest in only a few people. As of Series 10:
- Cameca ("The Aztecs"). The Tenth Doctor said in "Revolutions of Terror" (comic story) that he remained friends with Cameca & still visited her occasionally.
- Grace Holloway ("Doctor Who", the 1996 TV movie)
- Jabe ("The End of the World")
- Rose Tyler ("The Parting of the Ways", "The Impossible Planet", "Doomsday", "Journey's End", etc)
- Madame de Pompadour ("The Girl in the Fireplace")
- Astrid Peth ("Voyage of the Damned")
- River Song ("The Impossible Astronaut", "The Wedding of River Song", "The Husbands of River Song", etc)
- Queen Elizabeth I ("The Day of the Doctor")
However, the fact that he has shown attraction to many women so far does not exclude the possibility of also being attracted to people from other parts of the gender spectrum. We have not necessarily been shown all the relationships/attractions he has had, nor all the ones he will have.
Jack Harkness showed interest in both the Ninth & Tenth Doctors (in "The Doctor Dances" through to "The Parting of the Ways" & in "Journey's End"). The Ninth (more than the Tenth) seemed interested enough in this interest but, despite ample opportunity, the Doctor never persuade it. In "Smile", the Twelfth Doctor told Bill Potts, "I met an emperor made of algae once. He fancied me." Again, there was nothing to indicate that the Doctor returned this emperor's interest.
However, the Twelfth Doctor admitted that he had a crush on the Master when he was younger (in "World Enough and Time") which indicates that the Doctor is not straight. By human standards, he could be considered pansexual although that doesn't take into account that he's an alien that can be attracted to other aliens that are genderless.