My mom is 87 with advanced dementia. She wants to remarry an abusive husband. Do I have to be a guardian to prevent this?

If she has been diagosed with dementia nothing she signs is legal.

Does she live in a nursing home or memory care? Do you have an active power of attorney? Marriage is, in part, a legal transaction and if she's been declared incompetent and/or you have an active power of attorney then you just don't sign a marriage license.

If, on the other hand, she's living in someone's home and no legal things have been put in place she may be able to sign a marriage license without anyone knowing she shouldn't. In that case yes you may need to pursue guardianship, and/or an actual doctor's diagnosis to show she's incompetent.