Use tr to discard everything except tabs, and then count:
< input-file tr -dc \\t | wc -c
Bash uses a $'...' notation for specifying special characters:
$'...'
grep -c $'\t' foo
Use a perl regex (-P option) to grep tab characters.
-P
So, to count the number of tab characters in a file:
grep -o -P '\t' foo | wc -l