At boot, I am in 16bit protected mode after int 1589h.
The code selector in CS is 48 and is 16bit, 2GB size.
I am in the lower 64k. I want to jump over it, still remaining in 16bit mode. I am using USE16 directive.
If I use

jmp pword 48: 12345678h

it works, reloading CS with the same selector. But if I use

jmp dword 12345678h

it assembles fine but the program crashes.

use16 - directive to the fasm to create 16-bit code. It does not change the processor settings according to this code.

After calling the int 15h function 89h, a new selector from the newly installed GDT must be loaded into CS. The first command sets and, accordingly, loads new limits into the shadow part of the segment register. And the second one does not do this, but just tries to jump into nowhere.

ADD: Switch to PM of the processor, you do not need to call the int 15h function 89h. You might as well save the GDT value and load a new one.

The last jmp command loads an 8-byte descriptor from GDT (from memory) with an offset of 0x30 from its beginning into the shadow part of the segment register (in the CPU).

Further, the processor can already work with a segment longer than 64 KB. Such limits are set in the shadow part of the segment registers after switching on in real mode.

My bad, jmp dword didn't work simply because I had set a wrong ORG so the assembler would compute the wrong displacement for the relative jump.
Setting the right ORG fixes the 2nd jump.

By the way, I read in the Intel specification that EIP is loaded with 0000FFF0 at initialization. Is it guaranteed that at BIOS boot, the high 16 bits of EIP are 0 ?

Yes. Because there is no separate EIP, RIP or IP for different addressing modes. There is one register and a hardware mechanism that ensures that its value does not exceed the limits in the shadow part of the segment register. Of course, if the upper part contains nonzero bits, then you will get an #GP exception. In order to avoid spurious exception triggers when the protection mechanism (PE) is disabled, the higher part of RIP is maintained in the zero state. To set the upper four bits in real mode (RM), the value of the visible part of the segment register is used, which is set to segment 15 (0xF000) => 0xF000 * 0x10 + 0xFFF0 = 0xFFFF0