If my arms feel numb after weight lifting should I see a doctor?

Numbness signifies either compression of a nerve or decreased blood flow.  The most common scenario would be compression on a nerve either caused by inflammation after working out (like an adjacent tendon), or buy compression at the level that the nerves radiate out from the spine if the vertebrae are pulled together due to muscle contraction.  The nerves that supply the arms start off at the neck, so shrugs (for example) that work your trapezius muscles may tighten up your neck muscles and cause some compression.  You may also be tensing these muscles without noticing when you are doing certain presses or if you are using heavier weights and you are unable to isolate completely.
You should see an orthopedic who specialized in sports medicine in order to discuss your workout routine.  They can also observe the movements you make during your different sets and give you recommendations based on this.