Inspiron 570 not posting

These systems start as complete units in the $30 range -- so replacing a mainboard may make no sense.

Given that this system is headed toward 20 years old, you may want to consider upgrading to something much newer.

I know the pc is 16 years old but still I prefer older hardware I been using this pc since 2011 or 2012 I prefer older hardware it seems stable than newer hardware even thought I running a os that has out of support for 6 years now I just prefer older PCs from 2009-2015 I will try a different motherboard if that works I will buy the exact same motherboard from eBay next week

Before you buy replacement motherboard try replace CMOs battery and clear CMOs settings via motherboard RTCRST jumper.

smell plastic burn or smoke suggest a damaged component. Inspect the front and back side of original motherboard closely for black burn mark. When you suspect a focus of burn put your nose close to it if you don’t mind to sniff a vestige of burnt smell.  Replacement of mobo should cost no more than 10-15 usd in the us flea market

OP wish is completely personal preference 

second commentor happens to like old computer tech too, at the same time acknowledging often found free discarded old pc still working just needs some tinkering if that is some ppl hobby

Given that complete working systems are under $50, and that it will also net you a spare power supply and other components, consider that alternative.

Dell Inspiron 570 is a classic mini-tower desktop originally released around 2010. Designed for casual computing, it features AMD phenom II or Athlon II dual to quad core processors, DDR3 RAM, and a highly upgradable chassis. Today, these vintage units are often utilized as budget-friendly web-browsing machines or as beginner "sleeper PC" modification projects

I would say might also consider buying a Inspiron Intel based quad core motherboard/cpu combo as a slight upgrade of original AMD. 570 case can support standard mATX motherboard 

I replaced the cmos battery and remove and reinserted the jumper still doesn’t post still orange button and making fan noises and I know I could get a motherboard that has Intel but I am getting the exact same 04GJJT motherboard off eBay and the specs I’m using is a AMD Athlon II X2 245 3GB RAM 750GB/500GB/1TB HDD and Windows 7 Home Premium 32-bit and also dual boot Windows 11 Home 64-bit but I might go to Windows 10 Home 64-bit idk yet and when I get that new motherboard I am gonna do a clean installation of Windows 7 Home Premium 32-bit so that I can start fresh

I know I can go with a faster AMD cpu than that but I want to stick with the cpu I have and I know I can upgrade to 4-8GB RAM and upgrade to a SSD from a old 16 year old mechanical hard drive but I want to stick with all the stuff since it’s the exact same specs I used in 2014-2015 that’s why I have 2 Dell Inspiron 570’s one is family pc one is my main pc I play games on I also have a ASUS Radeon HD 6670 1GB GDDR5

And when I install Windows 7 again I am gonna install all drivers such as Chipset, Video, Audio, Ethernet and possibly update the BIOS since that motherboard I’m buying off eBay is probably gonna have a old BIOS version but it’s fine and I am gonna install the HD 4200 drivers first using 2009 Dell Drivers, Windows Update 2011 Drivers, and the latest Catalyst 2013 Drivers, and install the 2015 785G Drivers and when I install the HD 6670 I will install the Windows Update 2011 Drivers, ASUS 2011 Drivers, and latest Catalyst 2015 Drivers and do Windows Update via WSUS on Windows 7 RTM and via Legacy Update on Windows 7 SP1

Here are the pictures of the Motherboard

Note AMD oem board am3 socket yet using Intel square hole spacing cooler typical of Dell oem hybrid.  Looks like cpu fan is removed. 
sounds like OP plans to engage in a retro OS software install project.  

I did removed the cpu fan since I’m getting a nidec one from eBay when I tested the pc again I did plug everything back in including the 2 fans and the ram sticks and the cmos battery the pc still has orange button and makes fan noises

Although it's not very clear when zooming in, but from the picture you posted, it seems that capacitors bulging and leakage were the cause of failed motherboard.

How did you verify that the PSU wasn't the component that was smoking? Have you checked the capacitors on the motherboard?

If capacitors are blown you'll need a new motherboard. Dell doesn't manufacture parts for your system anymore, but you can try eBay or Parts-People. 

How I know the psu wasn’t smoking because I tried a different psu still didn’t work and the fans still spin on the psu so I knew it was the motherboard and when the pc is turned on it does high pitch fan noises and the button is on orange not blue

I got the new motherboard and a HD 5670 1GB GDDR5 and it works again reinstalling Windows 7 Home Premium 32-bit but now it beeps everytime after it posts I’m am running the diagnostics