Q: Will showing recent posts on my homepage cause a duplicate content issue?
A: As long as you don’t’ put completely duplicate content there is no problem , Even it is the case Search Engines  can easily figure out and disambiguate them .
And best practice here would be putting excerpt of the post instead of listing entire post contents.
Showing recent posts on the homepage also gives a refreshing feel to the home page and almost every Blogging Software’ and CMS’ are built like this nowadays.
Readers, Do you show recent blog posts …