So, there are a few angles to address on this.

The first and arguably most important is that the use of "romantic" here is problematic. While people of more varying class backgrounds hired domestic servants in the nineteenth century than do today, to the extent that some employers would be basically hiring the girl next door (a very common practice through the early nineteenth century in rural areas, and almost a form of apprenticeship for young women as future wives and mothers), in all cases there was an imbalance of power between "master" and servant. In most, it was great enough - being compounded not just by the fact that the servant, usually a young woman, was employed by the other party in question, but that she was employed by a person/family of higher social standing and income level - that the notion of free consent becomes hard to pin down. At the most extreme end, it's easy to see the power imbalance between a titled aristocrat with a large fortune, married and middle-aged, and a chambermaid or parlormaid who needs her job to earn her livelihood and can't risk turning him down, but even more moderate situations presented the same problem. A servant who was dismissed without a reference would find it very difficult to get a new job, because she would have no proof that she was experienced and trustworthy, so she was very dependent on her employer's goodwill.

On the other hand, apparently-romantic relationships did occur. While we need to keep in mind the obstacles to what we would consider a truly consenting and equal relationship today, we also need to be careful about writing our current understanding of consent over the experiences of women in the past who may not have perceived their situations as coercive. To a great extent, all women faced a significant imbalance of power when it came to accepting a man's romantic overtures, since marriage meant essentially giving him full control over her assets and her body, and she would have little recourse if he were abusive or controlling. Catherine Steedman's Master and Servant: Love and Labour in the English Industrial Age gets into the issue of dealing with these opposing viewpoints well in her chapter on, well, "Love".

The courting of maidservants was supposed to be done by men outside the household, with the male head of household often considered as a kind of surrogate father in the matter, particularly in the more informal, rural situations I described above, where there was somewhat less social space between employer and employee. It was the master's duty to see that she wasn't left abandoned and pregnant before she could make it to the altar. However, it did happen that relationships occurred between master and servant.

Much of the information on these cross-class relationships I'm aware of come from breach-of-promise cases described by Ginger Frost in Broken Promises: Courtship, Class, and Gender in Victorian England, which unfortunately is focused on the couples that didn't get married - if they did, they wouldn't have ended up in the courts, and they wouldn't have left much of a record of the issues they faced. Their breach of promise cases give us a number of details about servants' experiences of being in a relationship with their employers, and the problems that could arise. A big one is that an employer's family might object to his marrying someone of a lower social status, particularly if they had been living in the house while the servant (often a housekeeper) was working there. In some cases, the employer would ask her to wait for a father or uncle with some authority over him died, which could result in years of a secret engagement; with both people living under the same roof and the housekeeper acting as a kind of wife in terms of maintaining the household, it was not unheard of for them to end up in bed together and even for children to be born - premarital sex seems to have been comparatively common in these engagements. (Many employers also seem to have started sexual relationships with servants, allowing the latter to believe that marriage was on the table in order to obtain consent while never actually intending it.) Large age differences were also common in these couples - perhaps because it was often widowers who needed to hire a housekeeper - and sometimes this could lead men to rethink their decision to propose, in conjunction with other issues, since despite our modern idea that historical matches featured old men and young women, that was somewhat irregular.