I was just released from a major job in my field at the end of the probation period because of apparent performance issues (I was not made aware of them during my employment). It was my first full time job in a year as I was volunteering with a national service organization(but did not finish) and working in cafes. Should I include the three month job in my resume when applying for similar positions?
I wouldn't bother putting it on there.
The benefit of only three months experience is pretty marginal when looking over a resume - especially when you've only just entered the job market. On the other hand, you may end up with difficulties getting around the "performance problems" (whether they were justified or not).
You will be asked why you left - you could dodge that by falling onto the neutral "it turned out not to be a good fit" and you left at the end of probation.
Then you will be expected to provide a reference - and while you don't have to use that employer, it will be noted that you aren't using your most recent employer. If you do use that employer, there is a chance their side of the story won't mesh.
And if you do leave it off your resume, don't bring it up in the interview. You can use knowledge you picked up during those three months, but be careful about mentioning projects or teams.
End of the day - sorry that your first job didn't work out, and they didn't seem to see fit to guide you in your probation. You may well have dodged a bullet, though, because it may be an indicator of how they manage staff in general. On the plus side, you do have experience now (albeit, you don't want to document it), but you will have a better chance than many other candidates for your next applications. Good luck
Should I include the three month job in my resume when applying for similar positions?
Only if you can answer questions on it during the interview.
The purpose of the resume is to get you the interview. If you get that interview then you will have an opportunity to explain/answer any questions they have. To that end listing experience you have had in your field is typically beneficial to getting that interview with recent experience even more important if the job is in a field that is constantly changing/evolving.
The first five seconds of an interview is more important than any question they could ask about why you left your previous job, but you will need to be prepared to give confident and honest answers about it. If you had made a good impression in those first five seconds they likely will accept your answer. If you had not, then they will assume the worst.