Why did she burn his book? There's a reason that she's keeping him from Fallon
You're right, it didn't make sense since Liam proved that he wasn't a threat to the family or C-A, and she didn't have a reason to dislike him personally - in fact, this handsome heir of another dynasty was quite a catch.
But it also didn't make sense that Alexis suddenly felt the urge to "fix" her daughter's love life by encouraging Fallon's romantic feelings for Michael - and Alexis also didn't have a reason to like Michael
more than Liam.
But the Fallon/Michael romance could (and did) drive a wedge between Blake and his daughter since it's mostly Culhane who's screaming blue murder about the Clarke County scandal.
Alexis has a big plan, and the whole like/dislike is probably not one of her priorities. It looks like it's all about gaining controlling interest in C-A (via all of Tom's grandchildren, including the fake one) but the reason is still not very clear.
She came to back to get everything that once belonged to her, C-A is probably not one of those things.
However, Fallon (initially the power-hungry corporate bitch) may have her own plans and maybe she doesn't want to turn C-A in a Tom's Grandchildren Show, especially if she still hates Jeff.