The mushrooms growing out of your tree are produced by a wood rot fungus that is growing in the non-living wood inside the trunk. This type of fungus does not kill the live tissue in the outer part of the trunk, but it will rot out the interior portion of the trunk. Your tree may live for several years but the trunk may eventually weaken and break.
You might consider planting a young tree in another part of your yard as a future replacement for the older tree. Mushrooms are the fruiting portion of a fungus and removing them does not kill the fungus. There is no chemical treatment that will control the fungus without destroying the tree.
For more information see UC Pest Note 74109 – Wood Decay Fungi in Landscape Trees http://www.ipm.ucdavis.edu/PMG/PESTNOTES/pn74109.html
By V. Lazaneo, Urban Horticulture Advisor, Emeritus, UC Cooperative Extension, September 2012