Will SEO be affected if I change my theme?

As long as you don't introduce more dead links (404) into the store, it should be alright to change the theme. 

If some pages are going to be affected, set up redirects to preserve SEO juices. Google will recognize that the URL handles are redirected and you'll be fine.

Checking the new theme:

1. Make sure it's responsive. Support different devices. Test with Google Mobile-Friendly Test
2. Check if the page speed is fast. Check your score with Google Page Speed Insight.
3. Structured data (aka JSON LD) is set up by the theme (or you can use an app like SPO). Test with Google Rich Result Test.
4. If you are using an SEO app and you installed the code snippet to theme.liquid, reinstall the code snippet again
5. Sitemap.xml shouldn't be changed. But you can resubmit in Google Search Console (GSC) again. Just to have a quick test
6. Make sure you don't block site traffics with robots.txt
7. Make sure you turn off password mode so Google and visitors can visit your site. Some store owners just forget about it because they can access the store.

Monitoring your store:

1. Monitor the dead links after changing the theme with GSC. You can also do a real-time check with a dead link checker. It will scan all site links and show you which links are broken
2. Monitor search traffics with GA. See if your site traffics drop after changing the theme