Will keeping my cellphone under my pillow eventually give me head or neck cancer?

Not that we know. There has been no proven information that cancer is caused be cellphones. However why take the chance and don't put it under your pillow.
No. Most likely not. Radiation put out by cell phones have very limited penetration. Pillow is thick enough to degrade the radiation (distance and shielding). Long term use of cell phones next to ear may after many years of use introduce some risk, yet undefined. Specially in the very young children extensive use next to skin should be avoided.
No. I am very dubious about reports of cell phone related induced malignancy, but why do you need to keep the hand-held under the pillow?