The easiest way to redirect the user on the DAM tab is to override the default startupHelper used by Nuxeo by a custom one in you own module, something like:

import static org.jboss.seam.ScopeType.SESSION;

import org.jboss.seam.annotations.Begin;
import org.jboss.seam.annotations.In;
import org.jboss.seam.annotations.Install;
import org.jboss.seam.annotations.Name;
import org.jboss.seam.annotations.Scope;
import org.nuxeo.ecm.core.api.NuxeoPrincipal;
import org.nuxeo.ecm.platform.ui.web.api.WebActions;
import org.nuxeo.ecm.webapp.helpers.StartupHelper;

@Name("startupHelper")
@Scope(SESSION)
@Install(precedence = 11) // default install precedence APPLICATION is set to 10
public class CustomStartupHelper extends StartupHelper {

    private static final long serialVersionUID = -1L;

    @In(create = true)
    protected transient WebActions webActions;

    @Override
    @Begin(id = "#{conversationIdGenerator.nextMainConversationId}", join = true)
    public String initDomainAndFindStartupPage(String domainTitle, String viewId) {
        String result = super.initDomainAndFindStartupPage(domainTitle, viewId);
        NuxeoPrincipal principal = (NuxeoPrincipal) documentManager.getPrincipal();
        if (principal.isMemberOf("DAM Only")) {
            webActions.setCurrentTabIds("MAIN_TABS:dam");
            return "assets";
        } else {
            return result;
        }
    }
}

DM tab is a simple action, so you just have to override it. Here is the definition of this action:

<action id="documents" link="view_documents"
  label="label.main.tab.documents" order="40">
  <category>MAIN_TABS</category>
</action>

defined into the org.nuxeo.ecm.platform.actions component (don't forget to add the require item in your xml) So your contribution must something like:

 <component name="name.you.want.for.you.component">
 <require>org.nuxeo.ecm.platform.actions</require>
<extension target="org.nuxeo.ecm.platform.actions.ActionService"
    point="actions">
<action id="documents" link="view_documents"
  label="label.main.tab.documents" order="40">
  <category>MAIN_TABS</category>
  <filter id="yourFilterName">
    <rule grant="true">
      <condition>... put here your condition ...</condition>
    </rule>
  </filter>
</action>
</component>

Refer to the action documentation and filter documentation that explain With that you will limit the visibility of the tab.

About to not redirect to the DM view, may you will have information here.

Hope this will help you.

Thanks to Thomas and Bjalon for their quick response! I started with the code Thomas suggested, but to make it work I had to modify - mainly because the NuxeoPrincipal object is not available, nor is the CoreSession (documentManager) until Domain is initialized. The code below works as expected.

@Name("startupHelper")
@Scope(SESSION)
@Install(precedence = Install.APPLICATION + 1)
public class GripStartupHelper extends StartupHelper {

    private static final long serialVersionUID = -1L;

    @In(create = true)
    protected transient WebActions webActions;

    @Override
    @Begin(id = "#{conversationIdGenerator.nextMainConversationId}", join = true)
    public String initDomainAndFindStartupPage(String domainTitle, String viewId) {

        String result = super.initDomainAndFindStartupPage(domainTitle, viewId);

        NuxeoPrincipal principal = (NuxeoPrincipal) documentManager.getPrincipal();
        if (principal.isMemberOf("DAM Only")) {
            webActions.setCurrentTabIds("MAIN_TABS:dam");
            return "assets";
        } else {
            return result;
        }
    }

}