No, the BIOS cannot work without the CPU.
The very first thing your computer does when you power it on is checks that the connections between the CPU and RAM are good. This is a simple electrical check that all the bus line outputs are connected to the correct inputs. If this simple check doesn't pass, you get a beep code (or possibly an LED display on some motherboards). This check does not require the CPU, however without the CPU present, you may not even get beep codes, as even that requires some measure of processing (depends on the motherboard).
Once these things check out, however, then the CPU begins execution of the program contained within BIOS ROM that performs some additional higher-level checks (such as whether memory timings work, whether additional firmware for onboard devices can be loaded correctly, etc). This ROM program is written in x86 assembly language and does require the CPU to execute it.
What happened in your case is that the CPU is electrically compatible with the motherboard and does work, but lacks some low-level feature support that the motherboard depends on. Perhaps you used a CPU that has a higher TDP (generates more heat) than the motherboard can handle, has more cores than the BIOS knows how to initialize, or possibly doesn't support some power state the BIOS is trying to set. You didn't mention what CPU/motherboard combo you're using.
In whatever case, though, the CPU is providing enough support to the BIOS that it can execute the programs stored in its ROM (including moving bytes in and out of video memory). It just can't get further than that.
It's probably just checking the family and stepping IDs from the processor against an internal list of supported CPU types. In many cases, this can be patched with a BIOS upgrade. But you'd need to insert a supported CPU first in order to flash it.
13Obviously, the CPU is good enough to run the CPU check routine and display some text on the screen, but not good enough to boot an actual OS on your hardware. – Dmitry Grigoryev – 2018-01-15T08:32:04.827
22@DmitryGrigoryev "Obviously, the CPU is [...] not good enough to boot an actual OS on your hardware." Not necessarily so obvious. What if I want to boot, say, FreeDOS? What's the BIOS to have a say in what I boot on the PC? There's a reason why any MBR boot loader, even today, initially executes in real mode. – a CVn – 2018-01-15T08:37:45.290
7@MichaelKjörling I suppose the BIOS manufacturer decided that supporting TPM and making sure Windows users are happy is more important than letting you boot FreeDOS. – Dmitry Grigoryev – 2018-01-15T09:23:35.190
3@MichaelKjörling: The BIOS may hold microcode patches that fail to apply. There's no way to predict the usability of the CPU at that time. – MSalters – 2018-01-15T12:36:18.500
2What CPU is installed and in what motherboard? For example, did you somehow install a Coffee Lake processor on a Z270 or Z170 board? (This combination will not work because Coffee Lake and its Z370 chipset made incompatible changes to the socket pinout, despite using the same physical LGA1151 socket.) – bwDraco – 2018-01-15T16:50:23.223
Code in the BIOS is being run by the CPU (most likely using a very restricted subset which can be run on any modern x86 CPU). This code determines if the CPU qualifies according to rules also put in the BIOS. If the CPU fails, the message is shown. That the CPU is disqualified for full operation, does not mean that it cannot do anything at all. – Thorbjørn Ravn Andersen – 2018-01-16T10:05:04.927
@ThorbjørnRavnAndersen: Or maybe only CPUs that use the same physical socket as the motherboard. It might be unsupported for power delivery / stability reasons (or other physical / electrical reasons). But yes, limiting your code to a baseline below what the rest of the firmware assumes is the obvious mechanism. This is in terms of instruction-set, but also how the built-in graphics hardware is programmed, and stuff like that. – Peter Cordes – 2018-01-16T10:35:14.977
@PeterCordes Perhaps. Depends on the BIOS implementation. – Thorbjørn Ravn Andersen – 2018-01-16T10:37:11.940
@ThorbjørnRavnAndersen: Mobos usually need a BIOS update for a new CPU generation, e.g. a Sandybridge Z66 mobo could run an Ivybridge CPU, but only if you update the BIOS first. (By booting it with a SnB temporarily). Usually when a mobo is released, its initial BIOS supports all currently-existing CPUs that physically fit in the socket. (See also this comment where I wrote a better version of my previous comment.) New CPUs are the problem, not old.
– Peter Cordes – 2018-01-16T10:59:57.133