There is not really any safety issue with old forks, provided there is not a crack or advanced corrosion somewhere that could lead to loss of control... but this is rare. Do a careful visual inspection of the fork. Also if the front wheel seems to not track well or is not straight in the fork, do a more careful inspection as that could be indicative of such problems. Again, this is rare.
More likely the suspension will just stop providing any suspension, and TBH I am doubtful the forks that come on cheap bikes provide much benefit at the best of times.
Fatter tires at lower pressure are generally a more effective, lighter, and more reliable way to soften the ride than a suspension-shaped pogo stick on the front of your bike. Suspension forks that actually improve the ride are available starting around $400.