What happened between Lise and Ivan?
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.