Past tense (today)

"Today" covers the entire period of time from the beginning of the day to the end of the day. Therefore, parts of "today" can be in the past, parts of it can be in the present, and parts can even be in the future. The verb tense you use will reflect what part of "today" you're talking about:

Today, she is absent.

She is absent right now (e.g. "today's class/event/etc" is taking place right now, and she is not there).

Today, she was absent.

She was absent in the past part of today (e.g. it's currently the afternoon, and "today's class/event/etc" happened this morning (which is still part of "today"), and at that time she wasn't there).

Today, she will be absent.

She will be absent, at some time later today (e.g. it's currently the morning, and "today's class/event/etc" is going to happen in the afternoon, and at that time, she will not be there).