If you mean exemptions on your W-4, that would have the opposite effect.
Claiming 0 is as low as you can go.
If you claim 0 exceptions on your w4 and you still owe tax, you can have an additional amount withheld, or make "Estimated Tax Payments": quarterly.
If you mean claiming your exception on your tax return, claiming 1 exemption (your own) instead of 0 would normally result in a lower tax liability, but it all depends on whether you are a dependent to someone else or not.
