First, welcome to LSE, Josh. And congratulations on such a nice first question.
The answer to your question is No.
Inviting an officer into your house to talk about a robbery does not give the officer the right to search your house. Talking and searching involve two different amendments, and you must waive your rights under each explicitly. However, under the plain view doctrine, police who are lawfully in your house can search (and seize) your property without a warrant if:
a) the stuff they search or seize is in plain view (ie, “immediately apparent”); and,
b) they have probable cause to believe that whatever is in plain view contains evidence of a crime. (In other words, they can only search without a warrant if they could get a warrant, based on what is in plain view.)
In your hypothetical, the police do not meet all of the requirements for a warrantless search under the plain view doctrine. They are there legally, and the glass is in plain view. However, the DNA is not in plain view, which means they will need to search the glass to find it. To search the glass, they need probable cause to believe that your DNA will link you to a crime.
In this case, the need for probable cause creates a catch-22 for the police. Because they can't get probable cause merely from looking at the glass, the police must have had probable cause to get your DNA before walked in your door. But if they had probable cause before they showed up, then the 4th Amendment requires them to get a warrant. The plain view rule does not allow the police to skip the warrant requirement for their own convenience.
To see how seizing the glass would require probable cause, consider Arizona v. Hicks. In Hicks, police entered an apartment looking for a shooter. After finding the shooter, the police saw expensive stereo equipment that “seemed out of place in the squalid and otherwise ill-appointed four-room apartment.” Suspicious, they moved the stereo so they could read the serial number. When it turned out the stereo was stolen, they charged Hicks with theft. On appeal, the Court held that the serial number was not in plain view, and that seeming “out of place” did not establish probable cause to search the stereo. As a result, the search was unconstitutional.