Why a continuous supply of energy is needed to support a weight?
For skeletal muscles, you'll find a discussion in the Physics SE: https://physics.stackexchange.com/questions/1984/why-does-holding-something-up-cost-energy-while-no-work-is-being-done

To make things short: while holding a position, muscles need to maintain a given length. However, internally, the muscle looks like a collapsible array of filaments, with myosin heads acting as crossbridges between them. Even when the muscle is not contracting, myosin heads unbind and rebind at some rate and would let the muscle relax -- unless you input more energy to compensate this with a further contraction.

For smooth muscle, according to this article, the rate of unbinding can be very slow:


  Once the contraction has occurred, the cross bridge attachments need
  to be released in order that the muscle can relax. Although less is
  known about this phenomenon, it appears to be related to
  dephosphorylation of MLC20. In some situations, the dephosphory- lated
  MLC20 is very slow to allow detachment of the actin from the myosin
  cross bridge, resulting in a prolonged contraction. This has been
  referred to as a ‘latch-bridge’ (Hai and Murphy, 1988). This
  phenomenon may be of great value especially for tonically active SM
  beds as it would allow them to maintain basal tone through holding in
  an isometric state without a great energy cost.


So as the muscle is not relaxing, no input of energy is needed to adjust its length to the desired one. I assume a disadvantage of that is that the reaction time of the muscle is slower.