Canada has inherited the English common law, which allows you to sue for damages if a tort has been committed. I will focus on that, without considering the various other legal consequences of a wrongful prosecution (eg. criminal perjury charges, disciplinary action against police or prosecutors, ex gratia payments, or statutory remedies in Canadian legislation).
At common law, you can't sue a witness for giving false evidence. Witnesses are immune from civil liability for anything that they say in court, because otherwise the threat of litigation might deter them from giving truthful evidence. The rule was discussed by the Nova Scotia Court of Appeal in Elliott v Insurance Crime Prevention Bureau  NSCA 115. Cromwell JA said :
The immunity, of course, does not exist to protect wrongdoers, but it will sometimes do so. For the immunity to be effective, witnesses must be protected from all law suits, not only unmeritorious ones. This protection of witnesses from the risk of suit is seen as more important than righting a wrong in a particular case. Moreover, to achieve its objective, the immunity must be clear: People must know in advance whether they are protected or not ... The immunity, therefore, is a blunt instrument, barring all claims in the interests of the broader administration of justice.
As far as common law doctrines go, though, this area is a little unsettled: the Supreme Court of the United Kingdom limited the immunity for expert witnesses in Jones v Kaney  UKSC 13;  2 AC 398, departing from centuries of authority. But Jones v Kaney has not yet been adopted in Canada. See Walker v John Doe  BCSC 830.
Against a police officer or prosecutor, the ancient tort of malicious prosecution remains available in Canada. See Miazga v Kvello Estate  3 SCR 339. However, unfortunately for wrongly-prosecuted persons, it is difficult to establish because the plaintiff must prove that the prosecution was:
- initiated by the defendant;
- terminated in favour of the plaintiff;
- undertaken without reasonable and probable cause; and
- motivated by malice or a primary purpose other than that of carrying the law into effect.
In Miazga, the Supreme Court of Canada set aside a judgment for the plaintiff in a malicious prosecution action which had been affirmed by the Court of Appeal. The Supreme Court held that the courts below, having concluded that the prosecution was undertaken without reasonable cause, were wrong to conclude that the fourth element of malice had been established.