Is there any limit on how many times I can withdraw funds from my savings account every month?
As per Federal Reserve Board (Reserve Requirements for Depository Institutions Regulation D), usually you can withdraw money 6 times from your savings account in a month.
Your account may automatically be characterized as a checking account if you withdraw more than 6 times a month.
You must keep in mind that those 6 transactions will include pre-authorized automatic transfer, transfers via online banking (including bill pay), withdrawl by check, debit card, or similar order made by you and payable to any third party. If the bank sees you doing this more than 3 times, they may charge you a monetary penalty per transaction above the limit.
Apart from that, few banks just don't allow the transaction, and instead give an error message. Also, your account may automatically be re-characterized as a checking account.