It's possible that Lise wanted Ivan, but he rejected her. His reaction to her letter suggests this. However, he later says that he likes her and that what he said about her was a lie. So it's possible that they both have feelings for each other.
Madame Holokov mentions that...
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Ivan has come to visit Lise without her knowing. She says:
But forgive me, I can't trust my daughter so easily to your brother Ivan Fyodorovitch, though I still consider him the most chivalrous young man. But only fancy, he's been to see Lise and I knew nothing about it!
Alyosha is very surprised by this information. Madame Holokov says that Lise told her Ivan came to visit Madame and left quickly because she was asleep. However, soon after, Lise yells that she hates Ivan and tells Madame Holokov that she doesn't want him to visit anymore. Her mood swings are extremely severe.
When Alyosha visits Lise, she gives him a letter for Ivan. She says she'll poison herself if he doesn't deliver it. When he gives it to Ivan, Ivan recognizes the handwriting immediately. They discuss it, saying:
"Ah, from that little demon!" he laughed maliciously, and, without opening the envelope, he tore it into bits and threw it in the air. The bits were scattered by the wind.
"She's not sixteen yet, I believe, and already offering herself," he said contemptuously, striding along the street again.
"How do you mean, offering herself?" exclaimed Alyosha.
"As wanton women offer themselves, to be sure."
Later, though, Ivan says, "I like Lise. I said something nasty about her. It was a lie. I like her." It's impossible to know exactly what happened, but it does seem like the two characters have feelings for each other.