If your customers are paying by credit card, the restaurant where you work knows how much you are getting in tips, and they will report it to the IRS. So yes, you should keep track. Even if most of your customers do not pay by credit card, restaurants know the average tip for their business and will do the calculation and report it. So, for example, if you serve 10 tables, and each check is $50, the restaurant knows you did $500 worth of business, and they will report to the IRS that you made $75 in tips that night (15% of $500). Depending on the restaurant, they may use a higher or lower percentage (eg, when I worked at a pizza place near the college (college kids are poor tippers), they reported 12%). The tip amount for the year should show up on your W2 that you receive in January.
I waited tables all through college, and every place I worked reported the tips for me - and they took the taxes on those tips right out of those very low wages, so my paychecks were usually literally nothing.
ADDED: I would suggest that when your employer asks for your tickets at the end of the night, they are not only using that to make sure the numbers work out, they are also using that number to calculate your sales, and to approximate your tips based on both your sales and the credit card receipts. You should ask what percentage they are reporting for non-credit card sales.