After having a tooth extracted does the gum cause the other tooth in front to hurt?

Postoperative pain. Postoperative pain can certainly cause the teeth and gum tissue to hurt, but you may have an infection. See the dentist who did the oral surgery for advice.
Post op pain. If is normal to experience some pain in the gum tissue around the extraction site. Your dentist or oral surgeon may have reflected some gum tissue around the tooth to make it easier to remove. Follow the post op instructions and you should be fine. Warm salt water rinses will help as well as gently brushing the teeth near the extraction site.