Dallas here from the Shopify Social Care Team.
If you were to remove the tags from appearing on the blog page then they wouldn't be picked up by SEO either. I did talk with my themes team, though, to see if there's a way around this and they said that they'd be willing to make the color of your tags match the background color of your webpage so they won't be seen. Would you like me to get them to do that?
The other thing that you could look into would be to remove the tags entirely. That would only be if you don't want or need them for organization. So that the words are still used by SEO, you could then just add these keywords into the paragraph in the meta description under the Search Engine Listing Preview at the bottom of your blog page. This would essentially remove the use of tags, but would still have SEO pick up on these keywords.
Let me know if you want me to get our themes team to change the color of your tags.