I haven't noticed any perceptible latency from any active DP → HDMI I've used, and I'd honestly be surprised to see any
inexpensive active DP → HDMI adapter introducing latency, if only because doing so would require a frame buffer, which drives up costs.
What I have seen is DP → HDMI adapters that only support chroma subsampled pixel formats at 4kp60, and TVs that introduce many frame times' worth of latency due to various post-processing effects.
My hunch is as follows: given that consumer video devices (DVD/Blu-ray, set top box) almost exclusively output chroma-subsampled 4:2:2/4:2:0 YCbCr formats, TV post-processing pipelines may only support these formats, causing RGB (and possibly 4:4:4 YCbCr) signals to bypass post-processing, similar to the "PC" or "Game" modes present on some TVs that do the same.
In other words, if adapter → 4:2:2/4:2:0 and 4:2:2/4:2:0 → TV post-processing → latency, then adapter → latency, even if the adapter itself introduces no significant latency.
If I'm correct, the solution is an adapter which supports RGB output at the desired resolutions and frame rates and/or a TV with a PC/Game mode that bypasses post-processing for all source types, both of which are highly desirable for "monitor" use in any case.