I originally had 215/60R16 rims and tires on my vehicle. I have a set of nearly new 215/70R15 tires mounted on some 15 inch rims from another vehicle. If the lug bolts match up, can I replace them safely? If so, will there be any issues or concerns I should be aware of?
Short version: try it and see.
The 15" tires are 2.6% larger in overall diameter than the 16" tires. This will make your speedometer read 2.6% too high and effectively reduce acceleration of the vehicle by 2.6%. At 2.6% this is not really an issue for most people, e.g. when the speedometer indicates 70 mph you are really going 72 mph - not a big deal in most cases.
A bigger issue is probably the 15% taller sidewall on the 15" tire. Depending on how spirited your driving is, you may find that the 15" tires make the vehicle "wander" during, for example, lane changes, or the turns are less precise. 60 profile is common for SUVs which are somewhere between full size pickups and cars in handling, 70 profile is more of a full size pickup tire.
Another aspect to keep in mind is load carrying capacity - luckily you are keeping width the same and usually a taller sidewall means the load carrying capacity is higher, therefore you are probably safe, but check both tires' UTQG to be sure (https://en.wikipedia.org/wiki/Uniform_Tire_Quality_Grading).
Fitment of the tires may be different due to the wheels being different - since you have the wheels/tires already, easiest is to try to mount them and see if they fit. Usually front clearance is trickier than rear so start in the front. Mount 15" tires on one side and visually compare to the other side - do they look similar? This is a rough way to estimate offset, if you know the actual offset of both wheels you can also compare them. Does it look like the 15" tire is rubbing anything on either side (fender on the outside, suspension on the inside)? Spin the wheel after it is mounted to check for the wheel/tire hitting any parts on the car like brake calipers.