If your permalinks don't change then there shouldn't be much affect. Even if they do, as long as you use a redirect, the affect should be minimal. Just make sure your titles and meta tags remain in tact.

I agree with "Brian Alaway". Just don't change the permalinks

It kind of depends on which themes you are using.

(Examples: some might use H1s and H2s for post titles in certain areas and some are different. Some may have a lot of junk code behind the scene that interferes. Theme load times. Sponsor links (juice leaks)...etc)

But from one good theme to another, if the basic structure is the same, all your page/post links should stay the same. If you are using an SEO plugin, keep all the settings the same too and that info will transfer right along.

I recently switched themes on one of my sites and I have been noticing increased traffic and CTR, so sometimes theme settings and layouts can play a good roll.

Changing your theme can affect search results, because page load times and the quality of coding and architecture are becoming more important.

So if you go (for example) from a poorly coded free theme to a superbly coded premium theme you would immediately see faster load times and, over time, improved search results rank.

More details here:

Can Changing Your Theme Affect Your Search Result Placings? | WealthyDragon

Cheers,

Martin.