Do you have to update weight and balance charts after installing new avionics?
For the US under the FAA, Yes you do!

From the FAA WB handbook 


  Changes of fixed equipment may have a major effect upon the weight of
  the aircraft. Many aircraft are overloaded by the installation of
  extra radios or instruments. Fortunately, the replacement of older,
  heavy electronic equipment with newer, lighter types results in a
  weight reduction. This weight change, however helpful, can cause the
  CG to shift, which must be computed and annotated in the weight and
  balance record.


For larger repairs.


  Repairs and alterations are the major sources of weight changes. It is
  the responsibility of the FAA-certificate mechanic or repairman making
  any repair or alteration to know the weight and location of a change,
  to compute the CG, record the new empty weight and EWCG in the
  aircraft weight and balance record, and update the equipment lists.


Generally this is done by the shop that installs them. If you as the owner/operator installs them (which you are permitted to do to some extent) you may need to have a new WB drafted up. 

This is also checked at anual every year, 


  The FAA-certificated mechanic or repairman conducting an annual or
  condition inspection must ensure the weight and balance data in the
  aircraft records is current and accurate. It is the responsibility of
  the PIC to use the most current weight and balance data when operating
  the aircraft.