Does riding a bike with smaller wheels burn more or fewer calories than riding a bike with bigger wheels?

I track my exercise, and while I know that calories-burned counts are always an estimate, I'm wondering if I should mentally round up or down.

Let's say I ride for 45 minutes at 10 mph. Do I work harder or less hard on a bike with 20" wheels than I would riding for the same time at the same speed on a bike with 26" wheels?

Waaaay too many variables to track here. Tires, pressure, gearing used, weight of each bike, position of body on each bike, etc, etc. Get a HR monitor, do both rides, report your findings back here. I'd bet though, that on neutral ground, at 10mph the difference will be on the order of (diddly/squat) squared.

Use a heart rate monitor. You would likely want to have several rides on each bike for comparison to account for different weather and for differences in how you feel that day.

I started to use a formula at this link to track my exercise a couple weeks ago. I have no clue if it is accurate, but for my use to compare one workout to the next, it is good enough.
https://www.livestrong.com/article/30...ge-heart-rate/

Some heart rate monitors also have a built in calorie function. But I have a few monitors so I wanted to make sure that I use the same formula all of the time, as the different monitors use different proprietary formulae.

I think you will find that if the two bikes have the same tire pressure, that they both have very similar energy requirements. But different rider position and other factors will come into play. I for one would have a higher calorie ride if I was not as comfortable or if the bike geometry felt off, but that would be a geometry issue and not a tire diameter issue.