By and large, medical implants aren't made of significantly ferro-magnetic materials. Older generations of implants were typically made from stainless steel (which despite it's iron content is not ferromagnetic - you can use a magnet to test quality cutlery - if it sticks, it's garbage grade steel).
However, in some cases, even the mild ferromagnetism of stainless steel could cause problems - the classical example being brain aneurysm clips. These sit right on a fragile artery under the brain - even minor twisting in the magnetic field of the scanner can tear the artery causing immediate death.
In the case of orthopaedic implants, they are generally screwed into bones, and even if they are mildly ferro-magnetic, they're not going to go anywhere because the bones will hold them. What they will do, however, is cause degradation of the image in the immediate vicinity.
Non-medical foreign bodies, e.g. steel shrapnel, are a bit more of a problem. By and large, if a shrapnel fragment is small (less than 1 inch) and it's deep in muscle or skin then it's probably not going to cause a problem - however, if a fragment is near a vital organ, or in the eye - then all bets are off. People who are at risk of metal splinters in the eyes (metal workers, welders, machine shop workers, etc.) should have eye X-rays before MRI to make sure that there are no metal fragments which could move and damage the retina. In the case of ferro magnetic fragments, even if it's safe for MRI because it's held in by muscles, strongly ferromagnetic materials cause very severe degradation of the image. A guy once came for MRI of a knee, but had a 1/8" steel fragment lodged in his kneecap. It wasn't going anywhere, so the techs decided to try the scan anyway. The scan simply showed a great big hole from about half way down the thigh bone to about half way down the shin bone.
More modern materials like titanium, are completely non-ferromagnetic. So they are safe against movement, and titanium in particular causes very little image degradation (e.g. the image may be distorted for only 5 mm away from the titanium).
Metal of any kind, however, may get warm or hot - this is not due to the magnetic field, but due to the powerful radiotransmitter in the scanner (average transmit power during a scan may exceed 1 kW, with peak RF outputs of over 30 kW). Because metal may act as an antenna, it may get hot, so people with significant metal work will be instructed to tell the techs if they feel any sensation of heat, so that the scan can be paused, or a less RF intensive scan mode selected. This is also the reasoning behind the problems with tattoos - despite the hype, problems with tattoos are relatively uncommon. However, heating of tattoos is easily dealt with by covering the tattoo with an icepack or wet towel.
For larger ferrous objects, projectiles are a serious hazard. There have been several deaths due to people bringing oxygen bottles, or wheelchairs, into the scanner room. Alternatively, people have been trapped pinned against the scanner by gurneys or wheelchairs. In some cases, even 6 men were unable to pull the gurney off the scanner, and the scanner's magnet had to be 'quenched' to deactivate the field (Quenching an MRI is very impressive as 150 gallons of liquid helium violently boil and you get a huge eruption of white helium mist through the emergency vents - the problem is the $10k replacement cost for the helium, $20k in charges for the manufacturer to recommission the scanner, and the cost of 1-2 weeks of downtime. This assumes that neither the impact, nor the act of quenching, damages the scanner - both can easily cause a total loss).
Some people who have brought ferrous objects into MRI rooms have been caught out because the scanners are heavily magnetically shielded - usually with active shields. This means that the field strength suddenly rises like a brick wall as you get right up to the scanner opening - this is quite unlike more familiar magnets where the strength gradually increases as you bring them closer.