An extension to file is not an extension to pay tax.
No. An extension moves the filing deadline only. Tax owed is still due by the original due date, and interest and, in most cases, the failure-to-pay penalty continue to accrue on any unpaid balance.
For the standard federal extensions covered here - Form 4868, Form 7004, Form 8868 - no. They are automatic on timely, properly completed filing. A payment extension for estate tax under IRC §6161 is the exception: it requires showing reasonable cause.
It depends on the state. Some states honor the federal extension automatically, some require attaching a copy of it, some require their own separate form regardless, and some only require an extension filing if tax is due. See the master matrix for jurisdiction-by-jurisdiction rules as they are published.
Both the failure-to-file penalty (IRC §6651(a)(1), generally 5% per month, up to 25%) and the failure-to-pay penalty (IRC §6651(a)(2), generally 0.5% per month, up to 25%) can apply, along with interest under IRC §6601. See Penalties & Interest.
In limited cases. Taxpayers abroad who need more time to qualify for the foreign earned income exclusion can use Form 2350. Estates can request a discretionary payment extension under IRC §6161, or a longer deferral under IRC §6166 for a closely held business interest. See Special Situations.
It varies by state. Thirteen states plus DC impose an estate tax and four states impose an inheritance tax; most grant a filing extension, usually tied to the federal Form 4768 extension or a separate state form, and almost none extend the time to pay. Two states, Kentucky and Nebraska, have been confirmed against their governing statute to have no general filing-extension mechanism. Gift tax extensions are federal-only (Form 4768), with Connecticut the one state exception. See Estate & Inheritance Tax Extensions.
No. Form 720 (quarterly federal excise tax) and Form 2290 (heavy highway vehicle use tax) are not on the Form 7004 eligible-forms list, and neither form's own instructions provide an extension of time to file. See Federal Excise Tax Extensions.
Almost never. Sales tax is a trust-fund tax - the seller collects it from the customer and holds it for the state, so it was never the seller's own money. All 45 states that impose a general sales tax, plus DC, were checked: 33 report no general extension mechanism, and 13 grant one, almost always short, discretionary, request-based, and filing-only. See State Sales & Use Tax Extensions.
Federal payroll tax (Forms 941, 940) has no extension mechanism - neither form is on the Form 7004 list, and only a 10-additional-calendar-day filing accommodation exists for filers who deposited all taxes on time. State unemployment insurance (SUI/SUTA) quarterly reports follow a similar pattern in the states checked so far - no general extension mechanism, filed with the state workforce agency on a fixed quarterly cycle. FBAR (FinCEN Form 114) is the opposite case: it gets an automatic extension from April 15 to October 15 every year, with no request or form required. See Other Extensions.