It is never recommended that you plan trees over or near a septic tank. I would even be hesitant in even planting large shrubs like serviceberries (Amelanchier). Some individual may also caution you on the possible contamination human waste interacting with the fruit tree being a health risk to you but I personally think it's more of an issue if you're referring to the drain field.
You could plant shade loving plants like currants which should cause an issue to the septic tank unless you have to open the hatch for whatever reason.
I STRONGLY advise against it.
Even small trees can have extensive root systems. The roots grow in every direction and can get into every nook and cranny in an endless search for nutrients. Where nutrients are found, roots will grow larger to take advantage of the resource. The tiniest leak allows water, rich in nutrients, to seep into the soil. The roots find this, grow towards the feast. If the roots get into the pipe, they will grow, expanding the hole and expanding into the pipe. Roto-Rooter is one of many companies which are thriving as a result of tree roots growing into pipes and septic tanks. A simple cleanout is expensive enough, but only the existing root is removed. It'll grow back, giving you another chance to pay the plumber or a contractor for repairing major damage.
Seed the Mind, Harvest Ideas.
For what its worth. I had to dig up along side my concrete septic tank this fall because of broken pipes that were leaking. The top of the tank was approximately 2ft below the surface. The two leaking pipes were 3ft and 6ft below the surface. Approximately 40 years ago after the tank was installed my parents planted a white cedar (Thuja occidentalis) and a lilac (Syringa) over the edge of the tank. During the process of digging up the tank there were no unusual concentrations of roots on any feature of the tank that I could see or in the surrounding area. The majority of roots encountered were in the top 18" and were relatively small. A large tree may have caused problem. Expanding roots of perhaps a large poplar tree or something similar in size could certainly crack a concrete tank but these relatively small trees (cedar is 25 ft high and the lilac 10 ft) did not appear to be causing nor had the potential to cause problems in the future.
I am not making any recommendations just commenting on what I observed in this situation. Perhaps the potential of having to destroy a valuable food producing tree decades down the road to accomplish septic repairs is also a consideration.
Not to be the dissenting voice here, but there is more to this than just NO...
Trees do not crush septic tanks, lift sidewalks, crack rock or the related events they are so often contributed to doing. Trees, (99.9'% of them) like other botanicals, encapsulate what they grow over or on. They do very little penetration of a closed system...though some species can be more invasive than others....Willow for example can be very invasive and do a great deal of searching with there roots for water. When the find it, they can clog a system in short order if there is any leaks...leaks be the key focus...
Septic tanks tend to leak, so for that reason, planting trees around them is not advisable. Yet, a blanket No, if the tank is under 1 meter of soil there is less risk, and some folks will "restrict" the trees growth buy building a subsoil bin for them...Kinda like a giant bonsai...
This link is one of many good reads on the subject as all my old Arborist notes aren't digitized...
I would never plant anything near a septic tank or the drain field.
I have neighbor who has been in the business for over 50 years and his dad and grandpa built many of systems in the area well before that time. So I take him at his word. He says "Does it work? Do you like it working? Then leave it alone." Although he does say that technically they are supposed to be pretty well sealed and enclosed especially the newer they are so roots really shouldn't get in there, but that's a chance you take. A big chance and septic fixes can be really, really expensive.
I also would like to add that when I purchased my house the idiots before me planted a red maple tree a few feet from both the tank and line that runs out from the house. Nether are buried very deep and yet I've had no problems in the 13 years I've lived there. The tree is approx. 30-35 feet tall and as long as it keeps away it'll continue to live. The first problem and it'll be firewood and mushroom logs.
I've pondered if it's because of the shallow root system. ?
Elms have fine textured hairlike roots and are non invasive. I had several in the area of my septic tank at my previous home for 17 years and they are still there for over 20 years now with no problems.
Can you? Yes. It it a good idea? Not really. Do people get away with it? Yes. How much of a gambler are you?
There was a comment about possible contamination issues with fruit trees. Everything I have read says there is no hazard of the fruit being contaminated in this way.
Ben Falk was comfortable growing squash on his leach field and that is a higher risk proposition.
The main issues are not whether there might be contamination from the septic tank but the damage the tree roots will do to the tank and drain field. The drain field should not be driven on since compaction will damage the drains and any planting done should be with shallow rooted ground covers. That being said, the outside edges of the drain field are great for planting annuals.
We can green the world through random acts of planting.
My septic area gets the most sun of any area in my yard ... but I wouldn't plant anything that got deep roots near it for fear they would grow into the drain field. I grow shallow root plants in the ground on that side and use half submerged flower pots for annual flowers to add some color in early spring or growing herbs that I don't want to get out of hand. I have lots of my butterfly nectar and larval plants in that area and let things grow like desmodium and phyla that creep and spread, but don't have large root systems. Oh, and it's the favorite spot for spanish needles and the butterflies and bees love that and it's always the first to bloom for me in the spring here in florida. I'm planning on half submerging some solid pots this year, that will hold water, and try to grow water chestnuts and some other marsh food plants in them. It is frustrating to not be able to plant most things on the septic, but there are plenty of other things to choose from.