The phrasing "the rest of the ones" is grammatical, but sounds long and clumsy. Better options include:
- the rest
- the rest of them
- the others
- the other ones
As to your question of verbs, none of those feels correct or natural.
The problem is that the verbs at the end – "do", "get", "become" – are not the same as, or even compatible with, the main verb "be". By "compatible" I mean, it should be possible to join the verbs directly together, because there is an implied, omitted verb after the end of your sentence. The implied verb is the same verb written explicitly in the sentence: "be".
For example, your first sentence "This day will be forgotten just like the rest of the ones do" indicates that the omitted ending is "... just like the rest of the ones do be forgotten".
The verb "be" is fussy about which verbs are allowed to directly precede it. Although there are special cases where "do" is used with "be", such as commands like "Don't be silly" and "Do be quiet!", those are exceptional. When used without special meaning, "do be" feels completely wrong. That's why you can't use "do" here, even to end the sentence.
If the first verb was not "be", then ending with "do" could work:
- This day will pass just like the rest do.
The two other options you listed have the omitted endings "get be forgotten" and "become be forgotten", which also don't make sense.
Verbs which can safely go at the end of your sentence are (a) any tense of "be", or (b) a verb which can normally precede "be", such as a modal verb ("can", "could", "may", "might", "must", "shall", "should", "will", "would"), or "have [been]". It's also perfectly fine to omit the entire verb:
- just like the rest were.
- just like the rest are.
- just like the rest will be.
- just like the rest will.
- just like the rest have been.
- just like the rest have.
- just like the rest.
All this gives you a lot of possible combinations, but my preferred phrasing is the shortest:
- This day will be forgotten just like the rest.