TIPS are bonds. They have similar risks as other bonds. The most important risk in investing in TIPS is the interest rate risk.
Interest rate goes up. When interest rate goes up, bond prices go down. The interest rate goes up for two reasons: (1) inflation goes up; and (2) the real interest rate goes up. TIPS are not affected by the rise in inflation because their principal and interest are indexed to inflation. TIPS are affected by the changes in real interest rate. If real interest rate goes up after you bought TIPS, the value of your bonds goes down.
Low inflation or deflation. If inflation turns out to be lower than expected, the return from TIPS will be lower than that from regular bonds. Some people call this a risk, but I don't agree. Buying TIPS is like buying insurance. You insure against the risk of unexpected inflation. If you had fire insurance for many years and you never made a claim, you don't say your house not catching on fire is a risk or buying insurance was a waste of money. You had insurance protection for all those years. That's the value of the insurance. It's the same way with TIPS. Not having unexpected inflation is not a risk.
Your personal inflation rate may be higher than CPI changes. TIPS are indexed to the Consumer Price Index (CPI). Inflation for your own consumption will be different. You may buy more food, energy, health care or higher education than their weight in the CPI. As a result you may experience a higher rate of inflation than the CPI changes. TIPS will only protect you against CPI changes. There's no way to tie it to your personal inflation rate. But considering other investments like regular bonds don't even guarantee a rate of return above CPI changes, having some protection is still better than having no protection at all.
U.S. Dollar devaluation. TIPS are denominated in U.S. Dollars. If the U.S. Dollar devaluates against foreign currencies, your TIPS will buy fewer units of foreign goods or services. To the extent the price changes are included in the CPI, you are protected. If they are not, you have this risk. This risk is not limited to TIPS. It also affects regular bonds denominated in U.S. Dollars.
Taxes reduce inflation protection. Unfortunately you have to pay taxes on inflation. After taxes are paid, TIPS may not earn a rate of return above inflation. The higher the inflation rate, the lower the after-tax, after-inflation return. Regular Treasury bonds are affected by taxes as well. They may not have a positive after-tax, after-inflation return either. The higher the inflation rate, the lower their after-tax, after-inflation return.