If I replaced the fork on my cyclocross bike with a track fork, what would be the effect on the bike geometry?
Less fork offset on it's own means more trail. The fork length and wheel size will also affect this though, so you'd need to get all the numbers and maybe draw it out to work out the final difference in trail.

More (longer) trail makes the bike more stable (especially at speed) but less manoeuvrable. Touring and downhill bikes tend to have more trail, while city commuter bikes or sprinting/criterium bikes have less.