You don't necessarily have to be a "techie" in order to be able to manage a technical support team effectively. These guidelines explain how.

Recognise that output is more important to most people than input. In other words, what matters is not so much how long or how hard the technical support team works, but whether the systems are reliable and functioning well most of the time.
Most technical support problems have non-technical causes, and therefore non-technical solutions.
If you have just started in the role of managing a technical support team, undertake a fact-finding exercise, to determine what the technical support experience is for various groups of people in the school -- including the students. I have undertaken this work for schools on several occasions, and the findings often come as a surprise to the technical team.
Introduce reporting and measurement procedures. How many faults were reported this week? How long did it take, on average, to resolve them? What has been done to reduce the risk of the same fault occurring again?
Insist on the proactive involvement of the senior management team. In the work I've done with schools, a consistent message has come through that a passively supportive attitude, while better than an unsupportive one, is not enough.
Invite the network manager to your department or curriculum meetings, both to listen to what's important to you and, perhaps either briefly every time or, say, once every 6 weeks, to give a report about the network and any matters of concern.
If you are the educational technology co-ordinator or manager, work towards having the line management of the technical support team taken out of your hands. The technical infrastructure and support of the school ought to be regarded as a maintenance function, not part of a curriculum area.
In the meantime, allocate some of your budget for training purposes for the technical support staff, especially if they will be asked to implement or manage a new network system, say.
Ensure that there are clear guidelines for responsibility in place. The role of the technical support team is to advise, implement and maintain. It is your role to ensure that learning takes place. When new computer facilities are being planned, both parties will need to be fully involved in the discussions from the outset.