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sed "/$Title/d" BookDB.txt > BookDB1.txt && mv BookDB1.txt BookDB.txt
If you have GNU sed you can do an in place edit:
sed -i.bak "/$Title: $Author/d" BookDB.txt

You could test the return code from the grep

name="john"
age=25
file=txt
grep -x ${name}:${age} $file 2>&1 > /dev/null

if [ "$?" -eq "0" ]; then
echo "User exists"
fi

...