The above covers the heavy end of the tree.
Where I have problems, is carrying the small-branches upper half of a downed orchard tree on front forks. I've more than once torn a loader hydraulic hose. Once ripped off the oil pressure sensor wire while driving over debris to crush it. Going back to the barn with the red light on was unnerving, but essentially harmless.
When you get down to moving the small stuff, watch for risk to your hoses or worst case, the radiator, if debris carried up front starts to tumble off. Rasseling a downed tree with branches pointing in all directions requires more caution than you might expect.