Between zero and about a tenth of a millimetre of interference — the hole modelled slightly smaller than the pin — for a fit you press together once and expect to stay together.
That is a demanding target on an FDM machine, and whether you can hit it depends less on the number than on how repeatable your printer is.
Check your machine's spread first
A press fit needs a tolerance tighter than your machine's own variation. Run the numbers before designing one.
A 20 mm feature in PLA moves 0.0401 mm at the bottom of its contraction band and 0.1005 mm at the top. So material variation alone spans about six hundredths of a millimetre on a small feature — most of the interference budget, before the hole effects and flow calibration are counted.
In ABS the same 20 mm feature moves up to 0.1613 mm, which exceeds the entire budget on its own. Press fits in ABS need compensation or they need redesigning.
The technique that actually works
Print a test strip: the same pin diameter against a row of holes stepped in small increments. Assemble each one, note which is tight and which is loose, and use that number thereafter for that material and profile.
It takes one print and it replaces every rule of thumb, because it measures the combined effect of shrinkage, chord error, extrusion bulge and your own calibration in a single result.
Layer direction changes the answer
A pin printed standing up is round and weak across the layers. A pin printed lying down is stronger in bending and slightly oval, because its lower surface is flattened against the plate.
For a press fit, ovality matters more than strength — an oval pin in a round hole makes contact on two lines rather than a full circumference, which grips hard on insertion and loosens with use.
When to stop trying
If the joint has to be reliable, a printed press fit is usually the wrong mechanism. A heat-set insert, a printed boss with a machine screw, or a pin bonded with adhesive are all more repeatable and none of them depends on holding a hundredth of a millimetre.
Reserve press fits for things that can be reprinted cheaply if the fit is wrong.
Compute the contraction share for your own feature in the shrinkage calculator, and see fit classes in the tolerances and fit guide.