This is what we get:
You get this because footnotes are placed at the end of the document, no matter tabset or not.
---
title: "Example tabs"
output: html_document
---
## Title
Blah Blah
# {.tabset}
## Tab 1
Blah Blah ^[Footnote text for tab 1]
## Tab 2
More text ^[Footnote text for tab 2]
# Some other chapter
With text below the tabset

There is no easy solution to make it work otherwise as this is how footnotes are designed to be (at the end of a document).
You can try some Javascript code to move the footnote where you want them to be.
Or you could 'simulate' footnote in the tabs maybe .
---
title: "Example tabs"
output: html_document
---
## Title
Blah Blah
# {.tabset}
## Tab 1
Blah Blah^1^
1. Footnote text for tab 1
## Tab 2
More text^2^
2. Footnote text for tab 2