We don't know, and answering the question is complicated by the fact that the character doesn't follow standard "human" rules when it comes to that sort of thing, including never actually saying the phrase "I love you," which makes it hard to gauge. In the Modern Era the series has run several stories or story arcs in which the Doctor is shown in relationships that - if he were any other character - would qualify as romances. These include Rose Tyler, who the Doctor nearly said "I love you" to in "Doomsday" and spent the next two series pining over before being reunited in "The Stolen Earth"/"Journey's End"; Madame de Pompadour ("The Girl in the Fireplace"); Joan Redfern ("Human Nature"/"The Family of Blood" - it should be noted that the Doctor was under the influence of the chameleon arch which that temporarily rendered him human with no memory of being the Doctor during this time); River Song (whom the Doctor actually married in "The Wedding of River Song" and his romantic feelings for her were reiterated in "The Name of the Doctor" and "The Husbands of River Song"); and Clara Oswald. Clara is a rather complex case. In "Deep Breath" the Twelfth Doctor confirmed that, as the Eleventh Doctor, he thought of himself as Clara's boyfriend; Series 9 saw the Doctor exhibit in-love behaviour several times, including "catastrophizing" her potential departure and/or death, at one point stating outright to Clara that losing her would make it difficult for him to breath and he'd just run ("The Girl Who Died"), and after Clara does die, the Doctor spends 4 1/2 billion years on a desperate gamble to get her back, telling Ohila that the universe owed him this ("Face the Raven"; "Heaven Sent"; "Hell Bent"). When he erases his memories of her, dialogue states that he still remembers their adventures; he just no longer remembers anything related to her personality.
This only covers the TV series. The Doctor has been depicted as being in love with other characters in expanded universe media such as audio dramas and novels.