While there is no statutory law in British Columbia that requires that employees give notice of resignation, employees can be contractually obligated to give notice and even absent a contractual requirement there is a common law obligation in Canada that employees give notice. There's no fixed two week notice requirement, the amount of notice required depends on the circumstances. It would be necessary for an employer to prove damages that resulted from an employee quitting without giving notice or sufficient notice.
An example of this would be Consbec v Peter Walker where the defendant was determined to have failed to give reasonable notice of his resignation:
 Peter was obliged to provide Consbec with notice of his resignation. He did not give notice.
 The purpose of notice is to provide time for the employer to make
arrangements to have the work that the departing employee looked after by others,
or to find another employee.
However on appeal (2016 BCCA 114) the BC Court of Appeal determined that Consbec hadn't proven any damages had occurred as a result, so no damages were awarded:
 Although Consbec validly incurred costs and expenses of $5,875 as a result of Peter’s failure to give one month’s notice, it also saved $6,083 by not having to pay Peter’s salary during that period (i.e., $73,000/12). Accordingly, it suffered no damages as a result of Peter’s failure to give notice.
In many ways having a notice period specified in a contract is a much better option than ultimately leaving it up to a court decide. There seems to be very little precedent for this, especially for non-fiduciary employees. In Consbec the trial judge opted not to make a determination of what notice period was required in the case, something the appeal court determined was an error. The appeal court determined that a one month notice period was reasonable, but didn't explain how it came to this determination.
On the other hand, lawsuits like these a very rare for low ranking employees with little or no authority. As in the Consbec case, it would be hard for an employer prove damages. If Consbec hadn't been seeking large damages on other claims it's unlikely it would've brought a lawsuit.
Someone who tried to falsely claim to be sick in order to avoid having to work their remaining notice period would likely at minimum owe their employer the same damages as if they had failed to give notice. If the sick time is paid and/or includes benefits they'd have to pay back or compensate the the employer for these wages and benefits. In addition, because they're still employed, they still owe a duty of loyalty to their employers that former employees do not, and so could face additional damages for doing things like working for a competitor.
Someone who started work at another job couldn't use this as an excuse to avoid working during the required notice period. The person would have meet their obligations to both employers and both employers would be entitled to damages for the employee failing to do so. Again, the employee could face additional damages than those caused by their failure to work.