The Doctor married River Song and the fact he had romantic feelings for her was confirmed in "The Wedding of River Song", "The Name of the Doctor" and was a major plot point in "The Husbands of River Song".
If you wish to count one-off companions, the Doctor exhibited romantic feelings towards Grace Holloway ("Doctor Who"), Astrid Peth ("Voyage of the Damned") and Clara Oswin Oswald ("The Snowmen"), with the latter being considered a separate character from Clara Oswald.
The Doctor stated outright that he thought of himself as being modern-day Clara Oswald's boyfriend while in his Eleventh incarnation ("Deep Breath"), confirming what was indicated in "The Bells of Saint John", "Journey to the Centre of the TARDIS", "The Crimson Horror", "The Day of the Doctor" and "The Time of the Doctor", all of which included dialogue, actions and third-party discussion regarding his relationship with Clara as having romantic elements. The fact the Doctor still felt romantic feelings for Clara in his Twelfth incarnation were confirmed in "Time Heist" and "Dark Water" (in the case of the latter, the writer of the episode - also the showrunner and the creator of both Clara and the Twelfth Doctor- Steven Moffat confirmed that a line of dialogue the Doctor uttered: "Do you think I care so little for you that betraying me would make a difference" was explicitly meant to mean "I love you."
What makes things complex is the unwritten rule followed by the showrunners Russell T Davies and Steven Moffat that the Doctor does not say "I love you" to anyone - not even River Song (again this is actually a plot point in the 2015 Christmas special). Moffat, along with Peter Capaldi and Jenna Coleman, have all said the Doctor expresses his feelings for Clara (and others) in different ways. In "The Girl Who Died", the Doctor states that one day he'll miss Clara so much he won't be able to breathe, which is not a platonic statement. And his actions throughout "Heaven Sent" and "Hell Bent", coupled with Clara intentionally stopping him from saying anything approaching "I love you" in during their farewell in "Face the Raven" suggest there is more than a feeling of platonic friendship from the Doctor.
As for Classic Era characters, due to the rules set in place at the time no romance or hint of romance was ever allowed. There were some inklings with the Fourth Doctor and Romana, particularly in "City of Death", but nothing confirmed on screen (and generally any indications were actually signs of the real-life romance that blossomed between Tom Baker and Lalla Ward at that time). The 2006 episode "School Reunion" confirmed that Sarah Jane Smith had been in love with the Doctor, and it strongly implied that the Doctor was in love with her and other companions, but due to his longevity he generally refused to allow himself to become involved in that way.