Can I push a profile from Sandbox to Production?

Yes, pushing a profile from Sandbox to Production can be done. It is not possible to push through the Deployment Manager in Salesforce. The inbound/outbound change sets is still new and fairly limited. The meta-data for Profiles can be brought in using the Salesforce IDE. It is important that you bring down all elements that are included in the profile that you are pushing. Personally I just bring down everything that is available in the IDE meta-data in case I ever need it. Here are some simple steps to push the Profiles through the IDE.

1. Download and Install Eclipse IDE for Salesforce. You may find it here.
2. After setting up the IDE, create a "New Project" (File->New->Force.com Project
3. Give your project a name and description, enter your credentials to your sandbox.
4. When asked to select the metadata components, select the option for "Selected Metadata components" and click choose.
5. For ease of use, choose "Select All". This will bring down all metadata and may take some time to complete depending on the size of your org.
6. Click, "Ok" then "Finish"
7. You should now have all the elements that you need to deploy your profiles to production. You may do so by clicking on the profiles folder under the project to select it.
8. Once selected right click and select in the menu "Force.com->Deploy to Server"
9. Enter the proper credentials for your Production Environment and select the desired profiles from the menu.

That's really all there is to it. Ensure that you keep your IDE Project up to date and accurate so you don't push old settings by using the "Refresh from server" option often.