grep for a line in a file then remove the line

You could use sed,

sed -i '/test/d' example.txt

-i saves the changes made to that file. so you don't need to use a redirection operator.

-i[SUFFIX], --in-place[=SUFFIX]
             edit files in place (makes backup if SUFFIX supplied)

You're doing it the right way but use an && before the mv to make sure the grep succeeded or you'll zap your original file:

grep -F -v test example.txt > example.txt.tmp && mv example.txt.tmp example.txt

I also added the -F options since you said you want to remove a string, not a regexp.

You COULD use sed -i but then you need to worry about figuring out and/or escaping sed delimiters and sed does not support searching for strings so you'd need to try to escape every possible combination of regexp characters in your search string to try to make sed treat them as literal chars (a process you CANNOT automate due to the position-sensitive nature of regexp chars) and all it'd save you is manually naming your tmp file since sed uses one internally anyway.

Oh, one other option - you could use GNU awk 4.* with "inplace editing". It also uses a tmp file internally like sed does but it does support string operations so you don't need to try to escape RE metacharacters and it doesn't have delimiters as part of the syntax to worry about:

awk -i inplace -v rmv="test" '!index($0,rmv)' example.txt

Any grep/sed/awk solution will run the in blink of an eye on a 10,000 line file.