USPTO will remain open during the government shutdown. The patent office has indicated that it can operate for about four more weeks based on fees previously collected. After four weeks, the USPTO will begin shutting down per the shutdown plan described in the Department of Commerce plan.
Here is the excerpt in relation to the Patent Office on page 83:
The USPTO has sufficient funds from other than FY14 appropriations to continue full operations for approximately 4 weeks. Therefore, all 11,789 employees of the USPTO will be excepted for approximately up to 4 weeks following a lapse in appropriations. After this time period, the USPTO will execute the plan as described below.
The excepted employees will ensure the functionality of processes and systems minimally necessary for the preservation of patent rights, to allow compliance with statutory provisions that cannot be waived, and avoid disclosures of information that would be detrimental to the national security. Additionally, the excepted employees will ensure the functionality of processes and systems minimally necessary to preserve trademark rights. Preventing the public from accessing the USPTO’s electronic filing and payments systems may result in the complete or partial loss of intellectual property. Therefore, in order to prevent the loss of valuable intellectual property, these systems should remain open during any closure of the USPTO.