in no particular order...
Faculty - Finding a full-time teacher to coach the team isn't always easy, even when it's a paid position. The time devotion is a major drawback for many new teachers, who may be pursuing Masters or PhD, or being actively engaged in their social life. It's a lot easier to get older teachers involved, as many are already married and received all the degrees they wish to attain.
Resources - A FRC team needs a lot of stuff: lots of space for robots/parts/materials/tools, a lab or shop to build the robot in (preferably one with tools), access to computers for CAD/3D animation/website/etc, keys/security code access to the school for those late nights and early mornings during the build season, a place to drive the robot and assemble [mock-up] playing field components, and preferably a long-term storage space within the school.
Funding - This is a major issue. Whether it is justifying the funding to school districts looking to start teams, or trying to recruit sponsors and hash out a business plan. It's something that is constantly on the minds of nearly all FIRST teams, even the "established" ones.
Attitude - There are a lot of schools who are about only two things: meeting the minimum requirements of No Child Left Behind (and similar state measures) and sports/band. The best strategy here is to convince one administrator or teacher at a time, and slowly start building a coalition to support FIRST. This can take quite a while, and it needs perseverance, but tackling this attitude is part of the "changing-the-culture" aspect of FIRST.
Similar Competitions - They may already have an extensive pre-engineering curriculum, or participate in BotBall, Battlebots IQ, or other robotics competitions. In this case, this is one place where you don't really need FIRST, since they already "get it" when it comes to STEM.
Mentors - It can be hard to recruit good STEM mentors to your team if there are no good engineering/technical companies in the area. Parents also make great mentors, but some schools have strict rules on who can be involved with any school affialated activity.
School Rules - Anything from insurance reasons, to paying for janitors, to accessing the school during off hours, to attending out-of-state trips, to having non-school personal interacting with students in a school-sponsored team can all be nails in the coffin for potential teams.
Religion - You won't see many Amish FIRST teams.
