Originally Posted by WhyFi

You need a spacer to run a 10-speed cassette on an 11 speed freehub, regardless - you wouldn't be taking it off and putting it back on.

More importantly, the cog pitch, the distance between the centerline of the individual gears of the cassette, is slightly different between Shimano 10- and 11-speed - it's about 1/4mm narrower on the 11-speed. This means that you could fiddle with your indexing so that your current gear is smooth and noise-free, but with each successive gear change, up or down, it'll get more and more crunchy. So, realistically, you're not going to be sharing a cassette on the trainer between the two bikes, whether the cassette is 10-speed or 11-speed (unless you decide to upgrade at least the right shift/brake lever and rear derailleur to 105 on the Tiagra bike, which still seems the most sensible to me).

From a practical standpoint, you're also not going to share the trainer by swapping between 10- and 11-speed cassettes, as needed, either - it would be a total motivation-killer for whomever doesn't have their cassette on at the time. While it doesn't sound like a big obstacle, believe me, it would lead to a lot of skipped workouts, for one or both of you, over the course of the off-season.