Learn Hardcore Japanese: kyou ha muri wo sinaide kudasai.

〜ないで: “ <a class="border-b border-dotted border-blue-400 dark:border-blue-500 cursor-pointer text-inherit no-underline" href="/grammar/japanese/negation/zuni-zu-literary">without doing / don’t do” (for linking or requests)
〜なくて: often “and not/so not” or used for reasons/conditions (e.g., 困らなくてよかった “I’m glad I didn’t have trouble”), not for “please don’t.”

Breakdown of kyou wa muri o shinaide kudasai.

What does the particle は after 今日 do? Could I use another particle or drop it?

は after 今日 marks it as the topic: “as for today.” It can also sound slightly contrastive, like “at least today” or “today, specifically.”
Alternatives:
- 今日、無理をしないでください。 (no particle) — neutral “Today, please don’t…”
- 今日は無理をしないでください。 — topical/contrastive “As for today, please don’t…”
Don’t use 今日に here; に would mark a point in time for an action that occurs at/on that time, but topicalizing with は or just using 今日、 is natural in this kind of request.

Why is 無理 followed by を? Can I say 無理しないでください without を?

無理をする is a set phrase meaning “to overdo it / push yourself.” Here, を marks 無理 as the object of する.
Both forms are common:
- 無理をしないでください (slightly more formal/literary)
- 無理しないでください (a bit more colloquial)
Meaning is the same in everyday use.

What exactly is しないで doing here? How is it different from しなくて?