There has been romance between the Doctor & some, though not all, of his companions. For example, although the relationship between the Doctor & Rose Tyler wasn't a romantic one when it began, it did develop in that direction. This was especially obvious after the Doctor regenerated from the Ninth Doctor to the Tenth but romantic feelings had clearly been evident before then.
The most obviously romantic relationship has been that between several incarnations of the Doctor & River Song (Melody Pond), which resulted in their marriage ("The Wedding of River Song").
With Martha Jones, there were romantic feelings on her side -- for a time, at least -- although not on the Doctor's side.
With Donna Noble, there were no romantic feelings on either side; they were genuinely just close friends. Despite that, she & the Tenth Doctor were often mistaken for a couple.