Since packets travel over the wire have checksums on different layers, Ethernet and IPv4 have checksums for their headers, TCP's checksum even covers the entire segment.
I know it is not impossible that a corrupted packet, from the standpoint of the application layer, can slip in without being discarded by Ethernet/IP/TCP, because there are chances that their checksums are correct, only the probability is low.
I am designing a custom binary protocol for an IM application. My question is do I need to add a checksum to ensure the integrity of my application data? Is a checksum really needed in practice?
I would not bother with a checksum because of packets getting corrupted in the network.
However, since you are working on a protocol that would presumably be used on the open internet, you will need to prepare for rare cases of an unintended application sending udp packets or making tcp connections to your receiving/listening ports. Also there will be maybe less port scans and hackers / script kiddies knocking on your gates.
So you should make your protocol such that it is easy to discard this kind of traffic. Using a checksum in every transmission would imho be one sensible way of doing that.