The problem with teespring and the likes are that they have an MOQ that most people don't want to commit to.
As for hashtags and being irrelevant in a few minutes/days/weeks, I'd argue that a shirt that says "#GoBlue" will always be somewhat relevant. But hey, we'll see! More a fun project than anything else.
The MOQ is a drag but it serves a purpose: you can use a higher quality printing process (screen printing) which has most of the cost in the setup. Personally have not found too many Direct to Garment (what most one off shirts are printed with) printers that I was happy with in terms of consistent quality. Trust me, it would make my life so much easier if I could print quality one offs.
You may be better off doing each hashtag as a short term campaign, maybe Teespring has an API?
Love your idea though! Think you should consider different color shirts: when we did black shirts in our tshirt subscription, people complained that they already have too many black shirts in their closet and wanted something different.
As for hashtags and being irrelevant in a few minutes/days/weeks, I'd argue that a shirt that says "#GoBlue" will always be somewhat relevant. But hey, we'll see! More a fun project than anything else.