In brief: Domain Mapping breaks login, if blog is set to force user login before seeing blog contents.
Domain Mapping on subdomain-type WordPress network always redirects form subdomain to mapped domain. For example, if I have
one.page.com site, mapped to
one.com domain, whenever I enter
one.page.com URL, I'm redirected to
one.com domain, if Domain Mapping plugin is enabled.
If user has enabled (any plugin or own code) that user must be logged in in order to view frontend (blog contents) then login fails and user is all the time redirected back to login form, if Domain Plugin is activated.
This is problably because WordPress writes login cookie for
one.domain.com and this cookie is not available for
one.com domain after redirection done by Domain Mapping.
I don't know, if there is any fix for this problem, but maybe you try to read login cookie (if exists and if this is possible) from original subdomain before redirection and set the very same cookie for the domain you're about to redirect.