I've never used vellum in a printer myself, but here's a couple observations:
1. Make sure your printer is able to print to the size of paper you plan to use (check the manual). A lot of people don't know this, but you can't just put any size piece of paper in a printer or copier and have it work. Every copier, printer, and plain paper fax machine I know of has paper-detection sensors in the paper path that look for the paper at certain times during the copy/print cycle. If the paper is not there, or is there when it shouldn't be, the machine will jam and give a misfeed indication. Using odd-size paper, that the printer isn't 'programmed' to recognize, will cause certain misfeeds.
So, make sure the paper is
exactly 8 1/2 x 11" or 8 1/2 x 5 1/2" or whatever other compatible size you choose. (If you care, I used to be a copier/fax machine tech in a previous career and that's how I know this.)
2. Again, I've never used vellum myself, but I don't think it's likely the printer would "get too hot." Like plain-paper copiers, laser printers typically use what's called "fuser rollers" to melt and press the toner into the paper just before the paper exits the machine. The lower roller is often made of solid rubber, and the upper fuser roller is typically hollow steel with a teflon (or similar) coating. Inside the hollow portion of the upper roller is a "fuser lamp" which provides the heat. That heat is then monitored and regulated via a small temp sensor called a "thermistor," which sits up against the upper roller at all times (measuring the heat). When the machine is idle, the fuser lamp is off. When you go to print something, the lamp comes on and heats up the upper roller to the appropriate temp so it will melt the toner as the paper passes thru the fuser rollers (the lower roller is mainly to apply upward pressure from below). The thermistor communicates with the printer's main board (sort of a mobo for printers) and cycles the lamp on & off at the appropriate times -- i.e., turning it on when it's needed and turning it off when it's not.
Typically, there is also a "thermal fuse" (aka "thermofuse") sitting near the thermistor. If the temp should ever get too hot -- like, say, if the thermistor were to fail or something (or get too dirty) and keep the lamp on too long -- the thermal fuse will blow and shut the fuser lamp off so the machine doesn't catch on fire.
So, to answer your question, if the machine were to "get too hot" for some reason, it should shut down before any damage can occur. Since the paper you run thru the machine should have nothing to do with the temp of the roller, I wouldn't think vellum would make any difference.
You might check your owner's manual or call the mfgr about running vellum, as some printers have paper adjustment levers that either loosen or tighten paper guides accordingly when 'custom' paper is used (so the paper doesn't get jammed inside). Running thick card stock or construction paper, or really thin paper, would be examples of this.
Hope this helps. Best of luck with your wedding.
EDITED to fix a typo.