Early in Act I, when Nora and Mrs. Linde are speaking, Nora explains that Dr. Rank "is our greatest friend, and comes in at least once every day." The audience understands from this remark that Dr. Rank knows the Helmers well, and he chooses to see them often. Later in Act I, Dr. Rank arrives, and the audience hears him say to Torvald

No, my dear fellow, I won't disturb you; I would rather go in to your wife for a little while.

The frequency of Dr. Rank's visits, as well as his preference for Nora's company, suggests that he has deep feelings for Nora, but the reason as to why he has them is not clear, until a few moments later, after Krogstad leaves the scene.

After Krogstad leaves, Dr. Rank and Mrs. Linde speak of Mr. Krogstad's altered state, his diseased mind and moral corruption in general. Nora laughs off their serious talk, and according to the stage direction, she even claps her hands:

What do I care about tiresome Society? I am laughing at something quite different, something extremely amusing.

Nora's refusal to engage in the discussion might lead the audience to wonder at her odd timing and her blunt refusal to engage with a solemn conversation, but if Dr. Rank is in love with Nora, her flighty positivity might be what he likes about her. Even though Nora's joy, at this point, is a distraction to conceal her anxiety at Krogstad's visit, Dr. Rank may only see endearing girlish frivolity in Nora, who can, after all, be doll-like in her actions and appearances.

In Act II of Henrik Ibsen's A Doll's House, we find Nora is flirting innocently with Dr. Rank, the family friend, when, suddenly, he expresses her love for her. Although there are not many reasons that Dr. Rank can cite specifically, there is one hint that may lead us to understand his emotions. 

RANK:It is just that, that put me on the wrong track. You are a riddle to me. I have often thought that you would almost as soon be in my company as in Helmer's.

From these words, we could argue that Dr. Rank is the type of man that has lived a very conservative life, surrounded by science, and his medical profession, at all times. Notice how, when Dr. Rank explains his illness to Nora, he is bitter about the fact that his disease is common among people who tend to be debauched. Obviously, this is not his case, which is why he is upset that he has to pay for the "sins" of others.

RANK:Oh, it's a mere laughing matter, the whole thing. My poor innocent spine has to suffer for my father's youthful amusements.

Therefore, entering the home of Nora must be a new experience to Dr. Rank. He sees in her everything that he has never seen in a woman, or in anyone, before. Hence, he is able to appreciate, much more than Helmer, the extent to which Nora goes to try and please other people. He obviously cannot understand her personality. Then again, Nora is a riddle to understand, altogether. However, rather than diminish her to the role of a mere entertainer-the way that Helmer has- he actually does see between the lines, and he likes what he sees.