No, your webserver won't have to read the the whole body of the http request.
There is no necessity mentioned in the RFC
10.4.2 401 Unauthorized
The request requires user
authentication. The response MUST
include a WWW-Authenticate header
field (section 14.47) containing a
challenge applicable to the requested
resource. The client MAY repeat the
request with a suitable Authorization
header field (section 14.8). If the
request already included Authorization
credentials, then the 401 response
indicates that authorization has been
refused for those credentials. If the
401 response contains the same
challenge as the prior response, and
the user agent has already attempted
authentication at least once, then the
user SHOULD be presented the entity
that was given in the response, since
that entity might include relevant
diagnostic information. HTTP access
authentication is explained in "HTTP
Authentication: Basic and Digest
Access Authentication" [43].
Source: http://www.w3.org/Protocols/rfc2616/rfc2616-sec10.html