Hi DLEV,
Welcome to our Mattress Forum! 
It is not uncommon for more sensitive people to experience headaches from the VOCs emissions with a new memory foam mattress. Memory Foam tends to smell more and tends to have more chemicals than other foams or rubber/latex. Like with all when it comes to safety and toxicity it is a matter of degree whether the VOCs are harmful. Generally, I’d want at least CertiPUR-US certification for any foam used in a mattress or topper but even so, there is an increasing number of people that are more sensitive to certain materials, and their threshold for harmful substances and VOC’s is much lower than for the majority.
Like with everything else. it depends on the plastic or protector covering, its thickness, if it completely encloses the mattress or only partially, the level of VOCs emitted by the mattress oar topper and the Fire-Retardant solutions the particular mattress uses, the room temperature, and much more. One common issue with “plastic mattress covers” or protectors is that they emit large levels of VOCs themselves so it’s tricky to find one that does not. Out of all plastics used for mattress covering, there is a specially formulated polyethylene that is considered to be the safest and used for crib mattresses.
Here are a few thoughts that might help
~ If the product is returnable I’d consider getting a latex topper instead. It has similar pressure relieving qualities as Memory foam and is more responsive.
~ If keeping the topper is a must, I’d proceed to air it out for as long as possible during the day in front of an open window and door along with running a ventilation system.
~ An air filtration and purification system may also improve air quality but it has to have Activated Carbon filtration to absorb and remove VOC’s (HEPA filtering is not enough)
~ Generally, plastics are not breathable and this would create temperature-regulating issues and also may be quite noisy when you change positions during sleep. Any cover/protector you consider buying should be completely waterproof with heavy-duty seams and be thick enough to trap the VOC emissions.
~ One non-conventional solution would be to wrap the topper into an a large aluminum, polyethylene blanket with the aluminum facing down and seal it with tape. This may work for a while but would greatly detract from the topper’s comfort properties.
Again generally, the only reliable way to assess the “safety” of different materials in more general terms is based on lab tests and the certifications they have for harmful substances and VOCs so that you have some assurance that the VOCs are below the testing limits for the certification (see post #2 here for more information about some of the more reliable “safety” certifications). If the materials in a mattress or the mattress itself has a reliable “safety” certification then for most people they would certainly be “safe enough” … regardless of the type of material or the name of the manufacturer on the label.
Hopefully, that’s somehow helpful.
Phoenix