Most certainly not. Prayer is speaking to God, a willful act that engages the mind and thus can be done absolutely anywhere. The physical orientation of the body does not affect the validity of prayer.
Having said that, many religions have found that certain positions contain in themselves a facility for recollection or are more conducive for prayer; basically, certain positions help aid the attitude of prayer.
In the Catholic Church, during services, the faithful are prescribed to sit, stand and kneel for certain points throughout the ceremony. Often people are encouraged when at home to say their prayers kneeling, as it is a suppliant pose and emphasizes the humility one desires to achieve before God. It also can be uncomfortable, and thus a small penance can be offered up together with one's prayer. The ancient way of prayer was to stand, usually with arms slightly raised. The Jews often prayed thus, and it is retained in the Catholic Mass; often the priest assumes this stance at the altar during the canon. Prostration is used sometimes, but often appears dramatic and if one wishes to pray thus, it is best to do so in the privacy of one's own room.
The general rule is to pray in such a way as not to distract those around you, so following the local custom or maintaining a composed but reverent attitude is best. Ostentatiously moving your lips, nodding your head, making frequent movements with your arms, etc., are best left for home or when alone.