What are the drawbacks of transmitting RF frequencies through an antenna that is designed for different frequency?

1. Part of your output power will be ultimately wasted as heat in the antenna, feedline, matching elements, transmitter etc. (less range)
2. Energy will "bounce back" from the antenna to your transceiver. May potentially damage it.
3. Energy "bounced" from the antenna may cause intermodulation distortion in your transmitter. The transmitter will become a poor mixer (the device may not be compliant/legal, may transmit undesired frequencies).
4. When receiving (if the same antenna is used for RX/TX) you will "hear" other frequencies better than the one you are using for communication (worse sensitivity, higher noise level).

What happens if you send a broadband signal through a high pass filter? It gets attenuated. Same thing for an antenna more or less.

Here is a plot of what the impedance of a resonant antenna looks like. If you transmit signals in the capacative or inductive region, the current going into the antenna will 'see' more impedance and the signal will either reflect back to the receiver or get attenuated.

So that may be a good thing if you want that signal on that particular frequency to not get transmitted. Or a bad thing if you have a signal that needs to be transmitted with an antenna with the wrong impedance.

Source: radio electronics