Do police officers have to stop an interrogation when right to counsel has been invoked by a suspect or can they continue the questioning like nothing has happened?
Yes, for the jurisdiction specified in the tag
In the United States, questioning must stop as soon as the accused assserts his or her right to counsel. It's been upheld by subsequent precedent, but the origin is in Miranda v. Arizona, 384 U.S. 436, 474 (1966):
If the individual states that he wants an attorney, the interrogation must cease until an attorney is present. At that time, the individual must have an opportunity to confer with the attorney and to have him present during any subsequent questioning. If the individual cannot obtain an attorney and he indicates that he wants one before speaking to police, they must respect his decision to remain silent.
Under Miranda v. Arizona, interrogation must stop. Subsequent case law has refined when that rule holds. It is not sufficient to indicate a desire to speak to an attorney, it is necessary to unambiguously assert that right. Lower court rulings have held that it does not constitute a request for an attorney if you say that you would "like to have a lawyer" (Bane v. Indiana, 587 n.e. 2d 97), that you "felt that you might want" to talk to an attorney (Bunch v. VA, 304 s.e. 2d 271) or "I think I would like to talk to a lawyer (Clark v. Murphy, 331 F3d 1062). The matter was definitively settled in Davis v. US, 512 US 452, which held that
law enforcement officers may continue questioning until and unless a suspect clearly requests an attorney ...
if a reference is ambiguous or equivocal in that a reasonable officer in light of the circumstances would have understood only that the suspect might be invoking the right to counsel, Edwards does not require that officers stop questioning the suspect.
There is no reason to disturb the conclusion of the courts below that petitioner's remark-"Maybe I should talk to a lawyer"-was not a request for counsel
It has to be a direct assertion of that right, not just a polite request. On the other hand, for searches, it has been been deemed that all sorts of vague requests by police count as asking for permission to search ("Does the trunk open?", Schneckloth v. Bustamonte; "Do you mind if I check your person", US v. Drayton).
In England, you have an absolute right to have a solicitor (either your own, or a free duty solicitor) be produced "as soon as is practicable" on your request*. There is, however, no right to have the interview suspended at this point and the police have the power to continue questioning you until your solicitor arrives and then after you've received legal advice, although you can continue to invoke your right to privacy throughout.
Access to legal advice.
(1)A person arrested and held in custody in a police station or other premises shall be entitled, if he so requests, to consult a solicitor privately at any time.
(4)If a person makes such a request, he must be permitted to consult a solicitor as soon as is practicable except to the extent that delay is permitted by this section.
Anecdotally, the police will often stop the interview as soon as a solicitor is requested and the right to silence has been invoked as there's an assumption that further interrogation is basically useless.
**and no more than 36/48 hours in cases where your receiving legal advice "will lead to the alerting of other persons suspected of having committed such an offence but not yet arrested for it", etc.
Not really. There are court rulings stating that a suspect has the right to stop talking, but there isn't much in the way of actual recourse against the police other than the exclusionary rule. This rule requires that any information obtained from questioning the suspect after they have requested a lawyer not be admitted as evidence. The judiciary can only provided judicial relief; pressing criminal or civil charges against police who continue after a request for a lawyer would require legislative action to create a basis for such action.
So the police likely can continue to question the suspect, in the sense that there isn't anything really stopping them, but it will likely not be in the cops' interests, as any confession would be inadmissible, and there would "fruit of the poisonous tree" issues if further evidence is obtained, or can be alleged to have been obtained, from the questioning. However, the exclusionary rule is more limited in civil cases.
 Our colleagues LeBel and Fish JJ. also assert that our approach is such that the detainee is effectively forced to participate in the police investigation. The suggestion is that the questioning of a suspect, in and of itself, runs counter to the presumption of innocence and the protection against self-incrimination. This is clearly contrary to settled authority and practice. In our view, in defining the contours of the s. 7 right to silence and related Charter rights, consideration must be given not only to the protection of the rights of the accused but also to the societal interest in the investigation and solving of crimes. The police are charged with the duty to investigate alleged crimes and, in performing this duty, they necessarily have to make inquiries from relevant sources of information, including persons suspected of, or even charged with, committing the alleged crime. While the police must be respectful of an individual’s Charter rights, a rule that would require the police to automatically retreat upon a detainee stating that he or she has nothing to say, in our respectful view, would not strike the proper balance between the public interest in the investigation of crimes and the suspect’s interest in being left alone [emphasis mine].
As the law currently stands, arrested and detained individuals do not have the right to have a lawyer present while they are being questioned by police. Nor do the police have an obligation to stop questioning a detainee, even if he repeatedly and emphatically asserts his right to silence.
The 5th amendment jurisprudence has resulted in a rather complicated (painfully so to anyone who isn't a well-trained lawyer, and probably still so even then) set of rules of what happens in reaction to what. Here's an entire flowchart of it from lawcomic.net. I don't think it'll take long to gather what I mean by "painfully so" from that.
The short of the answer is as indicated in several other answers: in the simplest of cases, an unambiguous assertion that you are invoking your 5th amendment rights to remain silent and have an attorney is sufficient to end questioning, and cannot be used against you. However, follow the flow chart and you will see there are a lot of things that can resume questioning, can allow questioning to continue, or aren't protected by your assertion for counsel/silence. For example, if you unambiguously assert your right, and then go "you guys will never find where I buried him", then they can totally use that, and question you about it; you offered that of your own volition, and you can't start providing information and then arbitrarily stop. You also can't just sit there saying nothing; paradoxically, your right to remain silent requires you to speak up and unambiguously say you are exercising it.
If you've ever heard the advice to say nothing to a cop other than that you are asserting your 5th amendment rights to silence and counsel and you need your lawyer present for questioning, and then proceed to say nothing at all, the flow chart should help convince you as to why that is your best course of action.