Generally, plastic does not interfere with an RF signal. Metal is another story.
However, any plastic has a dielectric constant and can detune anantennaan antenna that is in close proximity. Detuning from proximity to a dielectric will happen mostly for plastic located close to a full size resonant antenna, and relatively little for a miniature (e.g. chip or ceramic) antenna.
Whether the effect is significant will depend on many factors, not least of which is how much performance you need from the antenna system. In other words, if you have enough margin in your application, even if you reduce your antenna efficiency by a few dBs you may not notice the difference.
Detuned antennas can generally be re-tuned by adding components in the antenna feed circuit, or adjusting their values. But avoiding trial-and-error by knowing exactly what measures to take requires expertise and use of specialized and expensive test equipment (particularly if, as in your case, you are talking about UHF or microwave frequencies).
If you are thinking of using a PCB antenna, you can try using a reference design, paying close attention to dimensions and PCB material properties. However, to be sure you've got it right, it may be best to enlist the services of an RF engineer to fine-tune it. This would be the case whether or not the antenna were boxed inside a plastic case.
As suggested earlier, ceramic or chip antennas are more self-contained and less likely to be detuned by nearby plastic, particularly if there is a bit of space between them and the plastic case. However, they cost more than a PCB trace and generally have less efficiency.
Avoid using plastic that is loaded with conductive material (metal, carbon to make it conductive) will reflect or absorb RF, or metallic paint.