Concrete form wood is often left in place in crawlspaces. According to the Washington State Department of Agriculture, who oversees Structural Pest Inspectors like myself, form wood left in crawlspaces is a conducive condition for Wood-Destroying Organisms (WDOs). If left in place, form wood can attract and harbor unwanted WDOs, which can then get into other parts of your home. For this reason, I always recommend any form wood be removed as completely as possible.
In some homes, removing form wood is a snap; they pop right off with a few taps of a hammer. In other homes, removing the form wood "as completely as possible" can sometimes mean "impossible." As in the photo of two wood form ties below, some form wood has concrete "spill-over," which would need to be broken off in order to remove the boards. And even in that case, getting 100% of the wood out from under the stem wall can be difficult to impossible to accomplish.
One solution that I've seen for stubborn form ties like these is to break them off at the stem wall and then cover the exposed ends of the form ties with new concrete. In this scenario, the concrete would act as a barrier to prevent any WDOs in the remaining wood from getting into the crawlspace.
It is also important to ensure that removing the form wood/ties leave no openings to the outside environment. If so, I suggest sealing the openings to prevent rodents and other unwanted pests from entering the crawlspace.
If you decide to leave the form ties in place (or just can't remove them), I suggest that any form wood remaining be regularly inspected for WDO activity.