You cannot effectively limit what clear evidence they find
Assume Bob owns a drug lab. He opens the door to the police and tells them, "I allow you to look for the lost cat of my neighbor." That does not limit the scope of what the police may find when doing a lawful search for whatever you specified to allow. During a reasonable search for that item you specified, it opens the door for any other evidence of a crime in plain view: the moment that they have legal and voluntary access to a location, and see the drug lab during their allowed search, any other such clear evidence is admissable under the Plain View Doctrine. Bob's rather sizable drug lab is not considered fruit of the poison tree, because it is in plain view once you enter the room, and he invited the police in.
That is, searching for different things results in different scrutiny in search. "Checking for injured people" has a much different reasonable scope than "looking for the nuclear code suitcase". As such, the item specified limits checking a room to places such a thing could be in, including (accessible) containers suitable for the item searched. You wouldn't hide someone injured in the kitchen drawer after all. But if the search reasonably brings the policeman to witness the clear evidence, then the item is most likely Plain View and admissible.
Note that it has to be clear evidence in plain view during a normal search for the items allowed to, not something that is in no reasonable connection or which needs to be carefully inspected to become evidence. If the search is for drugs, the police might reasonably open cupboards or paper bags (and probably find something clear like a bloody knife), but not folders and read the personal files or documents of the searched to spot tax fraud. The latter is by the way the pattern of United States v. Dichiarinte, 445 F.2d 126 (7th Cir. 1971):
Even if we assume that defendant consented to a search, the record shows that the consent was limited to a search for narcotics. The evidence at the suppression hearings contains repeated reference to the agents' interest in narcotics; and there was no indication that they desired to look for anything other than narcotics themselves. When defendant became aware that the agents were inspecting and seizing his personal papers, he attempted to call off the search. Under these circumstances, defendant's statement that the agents could "come over to the house and look" must be taken to mean at most that they might come and conduct only such a search as would be necessary to establish whether he had any narcotics. Government agents may not obtain consent to search on the representation that they intend to look only for certain specified items and subsequently use that consent as a license to conduct a general exploratory search.
[...]
Even assuming that these items were evidence of crime and thus subject to seizure, their criminal character was not apparent on a mere surface inspection, and defendant's limited consent did not authorize the agents' opening and reading them.
You can effectively limit where, how, and how long police searches
However, conditional consent can limit the search to specific rooms or parts thereof, explicitly exclude parts of rooms, or even contain a time limit (e.g. "You have 10 minutes"). Those limits have to be obeyed, and the consent of the search can be called off (see again, the cited part of United States v. Dichiarinte)
A typically unspoken condition is that searches are to be done without breaking things. For a locked container, police would need to either further inquire about keys, obtain consent to break the lock, or, of course, get a warrant. This limitation was ruled on in florida in State v. Wells, 539 So.2d 464, 14 Fla. L. Weekly 87 (Fla. 1989) (emphasis mine):
The Consent Search
The state urges us to hold that respondent's general consent to open and look into the trunk of the automobile was sufficient to authorize the opening of any locked or closed containers found there. In support of this argument, the state contends that Wargin correctly extended the principles of United States v. Ross, 456 U.S. 798, 102 S.Ct. 2157, 72 L.Ed.2d 572 (1982), to the consent-search context. We cannot agree.
[...]
In so holding, we decline to establish a rule that effectively would countenance breaking open a locked or sealed container solely because the police have permission to be in the place where that container is located, as in this instance. This would render the very act of locking or sealing the container meaningless and would utterly ignore a crucial concern underlying fourth amendment jurisprudence: the expectation of privacy reasonably manifested by an individual in his locked luggage, no matter where that luggage is located. 2
When the police are relying upon consent to conduct a warrantless search, they have no more authority than that reasonably conferred by the terms of the consent. If that consent does not convey permission to break open a locked or sealed container, it is unreasonable for the police to do so unless the search can be justified on some other basis.
As Jen noted, this is the backdrop for Florida v. Jimeno, where this stance was refined that it has to be a locked container, not just a closed paper bag.
Setting very strict limits, however, might become a reason for the police to instead seek a warrant to return, though on its own, not cooperating would not suffice to get a warrant.
Warrants are different.
If the police has a warrant, whatever you say about consent or not is irrelevant for the matter on the warrant: they may search for any item on the warrant in the manner prescribed on it, and if they find anything else that is clear evidence for a (different) crime, then that falls under Plain View Doctrine too.
