Will I be able to travel from Ireland to France if I'm negative for COVID-19 but don't have a vaccine?

I'm afraid crystal balls are in short supply these days. I'm not sure anyone can predict what, if anything, you will need to enter France from Ireland in 2 months.

It could range from "you can come as you wish" to "whatever your status you can't enter France without an imperious motive and you have to quarantine for 10 days", with "you need to be vaccinated" or "a negative PCR test is enough" in between.

The current situation is that you can enter France with just a recent PCR test (and a sworn declaration that you don't have symptoms, haven't been in contact with a confirmed case and that you accept a test on arrival), and vaccines actually currently don't exempt you from that requirement.

Current details are here: https://www.interieur.gouv.fr/Actualites/L-actu-du-Ministere/Certificate-of-international-travel#from2

The current plan is to allow fully vaccinated persons to enter without the PCR test.

Note that this can change very quickly, as it has happened multiple times, most recently with additional restrictions for the UK due to the spread of the Indian variant there.

For the last year and a half, it has almost always been possible to enter France from the EU. Since last summer, the only times when entry was effectively restricted is when France itself was under lockdown and domestic travel was also severly curtailed. Many other restrictions have applied to people coming from countries outside the EU but not on people travelling within the EU, and the government has signaled its intent to find ways to support the tourism industry and facilitates visits from abroad, even from outside the EU.

Save for a dramatic flare up of some new variant or something similar, it therefore seems unlikely that it would become mandatory to be vaccinated to enter France. The question would then become whether it would spare you the PCR test or any possible quarantine requirement but that's hard to tell with any certainty.