Before seeking a repair, you might want to do some troubleshooting of your own. Sometimes a runaway process can cause an iPhone to overheat. Chris Breen provides some tips in Troubleshooting a battery-sucking iPhone 4S. I solved a problem I had with my own overheating iPhone 4 by wiping the phone and reinstalling the OS. I did a fresh install, I didn't restore the phone from a backup. (This was when iOS 5 was first release, many of these problems have been sorted out now by OS updates.)
If you can't solve the problem yourself the next step is to take the phone to an authorized Apple repair center, in general an Apple Store. Which brings the warranty into play:
From Apple's (US) warranty for the iPhone 4
This warranty does not apply: (a) to consumable parts, such as
batteries or protective coatings that are designed to diminish over
time, unless failure has occurred due to a defect in materials or
workmanship; (b) to cosmetic damage, including but not limited to
scratches, dents and broken plastic on ports; (c) to damage caused by
use with another product; (d) to damage caused by accident, abuse,
misuse, liquid contact, fire, earthquake or other external cause; (e)
to damage caused by operating the Apple Product outside
They're not going to cover damage caused by misuse and they're not obligated to replace parts to fix cosmetic damage made by the user, though they will if its is a manufacturing or design fault. Your attitude when dealing with the Genius/Repair Tech may influence the outcome so I advocate being polite, pleasant and not acting like you are entitled to extra service.
If the damage is as you describe, I doubt you'll have any difficulties. In general, it is my experience that Apple tries to err on the side of the consumer. If the damage is worse than you describe and the repair tech feels that it in some way responsible for the overheating, then service may be refused. If that happens and you feel this is unfair, escalate to the store manager and Apple Customer Relations as you feel appropriate. You can reach Customer Relations through just about any Apple support line by asking for that department by name.
In all likelihood, you're making a mountain out of a mole hill. Take the phone in and have an Apple tech have a look. You'll most likely be pleasantly surprised.
See also, Gigaom: Apple Tech Support Tips: 4 Steps to Bend Apple to Your Will
As far as I know Apple stopped selling the 4s last fall. Therefore the person/reseller you bought it from might have twisted the truth. Where did you buy it? In the U.S.?
You can verify the warranty service and support coverage on: https://selfsolve.apple.com/agreementWarrantyDynamic.do
Apple Warranty starts on the date you buy it (date of purchase). Not on the day you start using it.
Apple:
WHAT IS COVERED BY THIS WARRANTY?
Apple Inc. of One Infinite Loop, Cupertino, California, U.S.A. 95014 (“Apple”) warrants the
Apple-branded iPhone, iPad or iPod hardware product and accessories
contained in the original packaging (“Apple Product”) against defects
in materials and workmanship when used normally in accordance with
Apple's published guidelines for a period of ONE (1) YEAR from the
date of original retail purchase by the end-user purchaser ("Warranty
Period"). Apple’s published guidelines include but are not limited to
information contained in technical specifications, user manuals and
service communications.
Source: US-Warranty http://www.apple.com/legal/warranty/products/ios-warranty-document-us.html
If you bought it at Apple and you are not happy with the sale, Apple usually gives 14 days to return the device.
If you do not need it I would suggest returning/selling the device or otherwise start using it. ;-)
Best Answer
If your iPhone is still under warranty then Apple will not reject a warranty repair unless they can determine that any damage is the cause of the issue.
Of course this will depend on how much damage there is to the phone, what the issue is, etc. Unfortunately your question doesn't specify what the actual issue is, and without a photo of your phone we don't really know what you mean by a small dent at the corner and small cracks on the screen.
Also, how do you know they didn't even check if the problem is caused by the dent or if it is manufacturing issues? In most cases there is no need to physically open the phone to determine if there is a fault.
If it was me I would be calling Apple directly and discussing it with them.