Using a foreign passport to travel in the USA will not by itself cause you to get stopped or deported. For example, US Border Patrol can stop you at an interior checkpoint regardless of any document you may or may not be carrying.
Similarly, if you are stopped, you will be liable to be deported, or not, regardless of whether you have a foreign passport in your possession. If you are a dual national of the US and another country, you are not deportable from the US. If you are a foreigner and deportable under 8 USC 1227, traveling without your passport will not protect you from deportation. If you are a foreigner who is not deportable under 8 USC 1227, traveling without your passport will not make you liable to deportation.
Using the foreign passport to pass through a TSA checkpoint could increase the chance that you would be investigated by a Border Patrol agent. As far as I'm aware, Border Patrol only monitors TSA checkpoints at a small number of airports that are within 100 miles of a land border, such as McAllen and Brownsville in Texas.
Border Patrol also sometimes investigates bus and train passengers within 100 miles of a land border. As with the highway checkpoints, these investigations typically start by asking the passenger about citizenship, so carrying the passport or not won't have much effect on the outcome of the investigation.
Except for the highway checkpoints and transportation hubs near the land borders, your chance of encountering Border Patrol is very small. Other officers are unlikely to investigate your immigration status.