Direct answer
It can. Direct HDMI paths and some matrix/extender systems add little perceptible delay, while video compression, scaling, frame-rate conversion, image processing, and network buffering can add latency. Even when the picture path is fast, television game mode, an AV processor, or projector processing may be the larger contributor.
For gaming locations, specify an acceptable end-to-end latency and test with the actual console, signal format, display, processor, and distribution route. If lowest latency is essential, a local console connection or bypass path may be preferable while other sources remain centralized. Do not rely only on a transmitter's advertised latency because the complete chain determines how controls feel.
How this answer was prepared
Camelback Smart Homes publishes this FAQ library for homeowners, design professionals, builders and property teams comparing integrated technology. Answers begin with the practical conclusion, then explain the factors that can change it.
Limitations: This is general educational information, not a site-specific design, quote, code determination or compatibility guarantee. Final requirements depend on the property, construction, infrastructure, equipment versions and project goals.