changing cassette

Almost certainly, yes.
That isn't a big change in the range that your rear mech has to deal with, and most derailleurs nowadays can accommodate up to 30t as standard. You *might* have to adjust the tension on the derailleur that holds the first jockey wheel close to the sprocket on the cassette, but even that's unlikely. If you are really unlucky you might need to put a couple of links in your chain as well to take the largest sprocket and largest chaining together, but again probably not.

I think you will need the 1mm spacer behind the 105 cassette, whereas I understand that the Tiagra cassette has protrusions behind the largest sprocket which act as the spacer.