Animals are optional for reproduction
While plants may employ animals to transport seeds, pollen, or other bodies carrying genetic material, fungi make use of other methods. These methods may employ animals, but would work perfectly fine without them. Additionally, most fungi are asexual - so there is no need to transfer genetic material with other "trees".
An equivalent to seeds, they are released in large numbers (sometimes in the trillions!), which increases the chance of survival. Spores are small, and are carried by air, so while contact with animals may spread them, it is really not necessary. Sometimes spores are also carried in a liquid as opposed to the traditional ~dust cloud
This occurs when part of the mycelium (root equivalent) or thallus (body) of a fungus breaks off unintentionally, then sprouts to form a new individual organism. Natural processes can easily break down your forest, but if you want animal contact, you can consider something that eats your "trees", maybe similar to a beaver - that spreads new organisms through its feces.
When cells divide off of a fungus for the specific purpose of reproduction (as opposed to tissues breaking unintentionally and sprouting), budding occurs. Animals cannot facilitate or aid in this process.
These methods will work fine independent of animal interaction, but you may employ animals to consume your "trees" if you wish.
Animals are needed to provide nutrients, but the type may vary
Why not just photosynthesize?
While your forest uses photosynthesis to produce sugar, which is an added boost, even plants use roots to obtain the nutrients they don't get through photosynthesis. Some things must come from other sources.
Why animals instead of other plants?
The problem with having a forest with set, unchanging flora is that the soil quickly loses the nutrients needed for healthy growth. That's why crop rotation is so important; many new nutrients must be introduced in some way. However, in nature, animal populations provide the solution when farmers aren't there to switch out what grows.
How should the forest take nutrients from animals?
The best way to constantly replenish your forest's nutrients is to kill some animals every once in a while. Luckily there are plenty of lifestyles that will allow for this:
- Carnivorous: Like carnivorous plants, your fungi may lure in animals such as insects and small birds, and break down their bodies to help them grow. There are plenty of ways to lure and trap animals - sweet scents, bright, attractive colors, closing parts, one-way pits, etc. The type of animal lured is not specific.
- Decomposing: Following traditional fungi, your forest could break down the corpses of (plants and) animals on the forest floor using its mycelium (roots). The type of animal decomposed is not specific. Additionally, feces may be broken down by the hyphae.
- Parasitic carnivores: You could consider growing your organisms on top of large living creatures. As long as the creatures are around, the forest will be. While this is appealing as it requires minimal work, it puts the "trees" in an evolutionary corner: if the hosts go extinct, so too do they.
- Parasitic decomposers: Your forest could intentionally kill animals, and break down their bodies. This varies from carnivores because animals are not being attracted actively. An example of this is releasing toxic spores that, when inhaled by living things, kill them, before taking root. Using their nutrients as fuel and building materials, the spores proliferate. The types of animals in these circumstances can vary.