The deadline for Tax Court petitions works like the deadline for tax returns: timely mailing is timely filing. You’re okay if you send your petition by first class mail on the last day for filing, which is normally 90 days after the IRS issues a notice of deficiency. For about three dollars extra you can cover yourself for a possible snafu by sending it by certified mail. There’s no need to use more expensive services, such as FedEx.
You can use FedEx if you want, but you have to make sure you use one of the services approved for this purpose. Those services are listed in IRS Notice 2004-83. They include (among others) FedEx 2 day, but not FedEx Express Saver, which is typically a few dollars cheaper (though still more expensive than certified mail). So when Marcius and Andrea Scaggs used Express Saver to send their Tax Court petition on the last day for filing, they failed to meet the deadline. The mistake may seem trivial, but the results are not: they permanently lost the ability to dispute their tax liability in Tax Court.
These taxpayers can still dispute their tax liability, but now they will have to (a) pay the tax in full, (b) file a claim for a refund, and (c) sue for the refund in federal court when the IRS denies their claim.