Tantalum capacitors do have a maximum peak current rating, and you should indeed worry about it (or rather, consider it in your design). Tantalums are notorious for becoming seriously damaged even when a max. voltage or current rating is violated only once, and only just briefly.
Example application guideline from AVX, KEMET and others will likely give you similar information:
1.2.4 Effect of surges
The solid Tantalum and OxiCap® capacitors have a limited ability to withstand voltage and current surges. This is in common with all other electrolytic capacitors and is due to the fact that they operate under very high electrical stress across the dielectric. For example a 6 volt tantalum capacitor has an Electrical Field of 167 kV/mm when operated at rated voltage. OxiCap® capacitors operate at electrical field significantly less than 167 kV/mm. [...] Solid tantalum capacitors and OxiCap® have a self healing ability provided by the Manganese Dioxide semiconducting layer used as the negative plate. However, this is limited in low impedance applications. In the case of low impedance circuits, the capacitor is likely to be stressed by current surges.
Derating the capacitor increases the reliability of the component. [...] In circuits which undergo rapid charge or discharge a protective resistor of 1Ω/V is recommended. If this is impossible, a derating factor of up to 70% should be used on tantalum capacitors. [...]
Also, the inductance of the wires between your source and the device's input may cause some ringing together with the input capacitors, which may lead to unexpectedly high voltages and a violation of the max. surge voltage rating. They tell you to use a derating of 70 % (!) for a reason when using tantalum caps.
Aluminum electroyltic capacitors take more abuse.