I am an EU citizen, and I don't need a visa for the United States. But there is an issue, as my visa is expired. If I continue to travel within the US, will they check my visa?
In general, no, your immigration status will not be checked whilst you are within the US.
However various organisations - most notably the Police - can check your immigration status at any time, and if they find your visa status is expired they can take steps up to and including detaining you and handing you to DHS (the Department of Homeland Security). This is very unlikely to occur unless you come to the attention of the police for other reasons, but it can happen.
By overstaying the time you are allowed to be in the US you are putting your chances of being allowed back into the US at any time in the future at risk. Once you've overstayed at all you are no longer able to use the Visa Waiver Program to re-enter the US in future, and will be required to obtain a visa. Depending on the length of time you overstay, you will likely be denied a visa for several years - the longer you overstay, the longer until you will be allowed back into the US again.