Here is a question from our clients-only extranet.
Continued payments are required by law for H-1 holder, unless they come under some very limited exceptions for leave for employee's personal reasons. Not paying, exposes the employer to investigation and penalties and may place the employee out of status.
In the green card context, non-payment can lead to problems with demonstrating ability to pay wages.
For both H-1 and GC, nonpayment can lead to an assumption that no genuine job exists. That could lead to cancellation of one or both processes, except for situations where AC21 portability is involved.