A government shutdown mainly just affects discretionary spending and the important parts of the Affordable Care Act are mandatory spending or would still be able to operated. From the CRS:
It would appear that substantial ACA implementation might continue during the lapse in annual appropriations that resulted in a temporary government shutdown for two reasons. The first is that the federal government will be able to rely on sources of funding other than annual discretionary appropriations to support implementation activities, including multiple-year and no-year discretionary funds still available for obligations as we as mandatory funds. The second reason, which is covered in more detail in answers to several of the other questions, is that agencies may continue to perform certain types of activities that fall under exceptions to Antideficiency Act, which generally prohibits continued operations in the absence of appropriations.
So even if Lee had the votes to shut down the government, which he doesn’t, the implementations of Obamacare would be one of the government functions that would keep going anyway. While much of the government would crawl to a halt, the ACA would not.
This plan would only work if Obama was willing to gut his own signature law to prevent this temporarily issue and that is never going to happen.
What Lee is doing is pure political grandstanding. His vote wasn’t needed for the CR and he was probably never going to vote for it anyway. This way he can get love from his base for fighting tho good fight, even though it is really just a doomed piece of political theater.