Can a receiver get damaged if you power it up without an antenna?

Running a receiver without an antenna should not cause any damage to the receiver, as no RF power is sent to the antenna (if your receiver has an active antenna there may be a DC bias present, but this does not cause problems - you will often see this in GPS receivers.)

However, for completeness, it is worth noting that some modern RC receivers actually do transmit, to send telemetry and other data back to the controller; in this case 'receiver' is a misnomer, used to mean "the radio part in the aircraft" - and it is possible for damage to occur. The correct term would be 'transceiver', a portmanteau of 'transmitter' and 'receiver'.

The reason a transmitter can be damaged when run without an antenna is because the power sent to the antenna has nowhere to go. When a (matched) antenna is connected, the power is transferred to the antenna but without it the energy is reflected back down the transmission line to the output transistors. These transistors are not designed to handle that much power coming back, and can fail. The likelihood of this happening generally increases with transmit power, but it also depends on the rating of the output transistors so this is just a crude approximation.
(Transmission line theory deviates somewhat from the normal models of power transfer, where having nowhere for the power to go would create an open circuit, and no current flows.)