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 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.
We have to infer about what exactly happened between Lise and Ivan, but there is plenty of information to go on.
The most likely best answer to this question is simply that Lise "falls for Ivan" and Ivan is not interested.
Lise almost certainly "offered herself" in marriage to Ivan, as Ivan tells Alyosha. There is some reason to wonder if Ivan is being sarcastic when he tells Alyosha that Lise has offered herself to him, but he does say this.
Alyosha asks, "Offered herself how?" and Ivan replies that she offered herself as "young girls do". We are left to wonder at the precise meaning of Ivan's words, but the dynamic is clear. Lise is prepared to choose Ivan over Alyosha and in fact attempts to do so.
The letter she writes and asks Alyosha to give to Ivan is most likely a letter describing Lise's willingness to do exactly this. Ivan tears up the letter without reading it, so we can't be completely sure what it said. Again we must infer and trust the gist of what Ivan has told Alyosha.
To remove any doubt about the situation, we should point out that the two, Lise and Ivan, spend some time together when Ivan comse back into town after Dmitri's arrest, but aside from a brief chat during which Lise apparently declares her love for Ivan, it seems nothing "happens" between them.