It is legal to hunt mushrooms on all state owned lands. This includes state parks, state forests, state fish & wildlife areas, and state recreational areas. The only restrictions are in nature preserves. You are allowed to hunt mushrooms if the nature preserve is a part of a state forest, state park, etc. If it is a standalone property, then the nature preserve makes their own rules.
The only legal question is whether you can leave established trails to hunt mushrooms. The Indiana code was updated in 2014 to allow activities that are licensed (such as hunting deer) or exempt from licensure (such as mushroom hunting) to leave the marked trails while in pursuit of that activity. A short writeup with the relevant information can be found here.
It is legal to hunt mushrooms on any property in the Hoosier National Forest, as long as they are for personal use. Harvesting for commercial purposes is not allowed. Rules link
Hunting in local city parks is up to that individual community parks department.
Indiana Dunes National Lakeshore
Mushroom collecting is not allowed anywhere on the property. Collecting is allowed at the Indiana Dunes State Park.
St Joseph County Parks
"Mushroom hunting and picking of wildflowers is prohibited in all of the St. Joseph County Parks." Rules Link
ACRES Land Trust
These properties are located in the NE section of the state. They are available for hiking, birding, photography, etc., but you must stay on the trails, and you cannot collect any mushrooms. They offer permits for researchers, but you still cannot collect, even with a permit. ACRES Land Trust
Sycamore Land Trust
These properties are located around the Bloomington area. They are available for hiking, birding, photography, etc., but you must stay on the trails, and you cannot collect any mushrooms. They do offer permits for researchers. Sycamore Land Trust
NICHES Land Trust
NICHES Land Trust runs 30 properties totaling 2,638 acres. We have permission to hunt mushrooms on these properties. NICHES serves 13 counties in the Lafayette area. You can view NICHES properties on our map of hunting locations. If you do hunt these properties, please take pictures or otherwise document the species that you see. The organization has requested these lists if they are available. Send pictures/species lists to firstname.lastname@example.org
Section of Indiana code allowing mushroom hunting on all DNR properties:
312 IAC 8-2-10 Preservation of habitat and natural and cultural resources
Authority: IC 14-10-2-4; IC 14-11-2-1
Affected: IC 14
Sec. 10. Except as authorized by a license, a person must not do any of the following within a DNR property:
(1) Molest an animal den or bird nest.
(2) Collect any wild animal, except as authorized by section 3 or 4 of this rule.
(3) Damage or collect a plant or pick flowers. Exempted from this subdivision are the following: (A) Berries.
(D) Fallen cones.
(G) Greens. (4) Pick berries, fruits, nuts, fallen cones, mushrooms, leaves, or greens on a nature preserve, or another property administered by the division of nature preserves, unless the property is located at any of the following: (A) A state park. (B) A reservoir property (C) A fish and wildlife area. (D) A state forest.