Same problem here. Here is the log file:
DKMS make.log for r8168-8.049.02 for kernel 5.18.10-76051810-generic (x86_64)
Fri 09 Sep 2022 12:38:04 PM CDT
make: Entering directory '/usr/src/linux-headers-5.18.10-76051810-generic'
  CC [M]  /var/lib/dkms/r8168/8.049.02/build/r8168_n.o
  CC [M]  /var/lib/dkms/r8168/8.049.02/build/r8168_asf.o
  CC [M]  /var/lib/dkms/r8168/8.049.02/build/rtl_eeprom.o
  CC [M]  /var/lib/dkms/r8168/8.049.02/build/rtltool.o
/var/lib/dkms/r8168/8.049.02/build/r8168_n.c: In function ‘rtl8168_mac_loopback_test’:
/var/lib/dkms/r8168/8.049.02/build/r8168_n.c:3668:17: error: implicit declaration of function ‘pci_dma_sync_single_for_device’; did you mean ‘dma_sync_single_for_device’? [-Werror=implicit-function-declaration]
 3668 |                 pci_dma_sync_single_for_device(tp->pci_dev,
      |                 ^~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
      |                 dma_sync_single_for_device
/var/lib/dkms/r8168/8.049.02/build/r8168_n.c: In function ‘rtl8168_get_mac_address’:
/var/lib/dkms/r8168/8.049.02/build/r8168_n.c:24140:34: error: assignment of read-only location ‘*(dev->dev_addr + (sizetype)i)’
24140 |                 dev->dev_addr[i] = RTL_R8(tp, MAC0 + i);
      |                                  ^
In file included from ./include/linux/string.h:253,
                 from ./include/linux/bitmap.h:11,
                 from ./include/linux/cpumask.h:12,
                 from ./arch/x86/include/asm/cpumask.h:5,
                 from ./arch/x86/include/asm/msr.h:11,
                 from ./arch/x86/include/asm/processor.h:22,
                 from ./arch/x86/include/asm/timex.h:5,
                 from ./include/linux/timex.h:67,
                 from ./include/linux/time32.h:13,
                 from ./include/linux/time.h:60,
                 from ./include/linux/stat.h:19,
                 from ./include/linux/module.h:13,
                 from /var/lib/dkms/r8168/8.049.02/build/r8168_n.c:43:
/var/lib/dkms/r8168/8.049.02/build/r8168_n.c: In function ‘rtl8168_set_mac_address’:
/var/lib/dkms/r8168/8.049.02/build/r8168_n.c:24171:19: warning: passing argument 1 of ‘__builtin_memcpy’ discards ‘const’ qualifier from pointer target type [-Wdiscarded-qualifiers]
24171 |         memcpy(dev->dev_addr, addr->sa_data, dev->addr_len);
      |                ~~~^~~~~~~~~~
./include/linux/fortify-string.h:362:27: note: in definition of macro ‘__fortify_memcpy_chk’
  362 |         __underlying_##op(p, q, __fortify_size);                        \
      |                           ^
/var/lib/dkms/r8168/8.049.02/build/r8168_n.c:24171:9: note: in expansion of macro ‘memcpy’
24171 |         memcpy(dev->dev_addr, addr->sa_data, dev->addr_len);
      |         ^~~~~~
/var/lib/dkms/r8168/8.049.02/build/r8168_n.c:24171:19: note: expected ‘void *’ but argument is of type ‘const unsigned char *’
24171 |         memcpy(dev->dev_addr, addr->sa_data, dev->addr_len);
      |                ~~~^~~~~~~~~~
./include/linux/fortify-string.h:362:27: note: in definition of macro ‘__fortify_memcpy_chk’
  362 |         __underlying_##op(p, q, __fortify_size);                        \
      |                           ^
/var/lib/dkms/r8168/8.049.02/build/r8168_n.c:24171:9: note: in expansion of macro ‘memcpy’
24171 |         memcpy(dev->dev_addr, addr->sa_data, dev->addr_len);
      |         ^~~~~~
/var/lib/dkms/r8168/8.049.02/build/r8168_n.c:24173:32: warning: passing argument 2 of ‘rtl8168_rar_set’ discards ‘const’ qualifier from pointer target type [-Wdiscarded-qualifiers]
24173 |         rtl8168_rar_set(tp, dev->dev_addr);
      |                             ~~~^~~~~~~~~~
/var/lib/dkms/r8168/8.049.02/build/r8168_n.c:566:59: note: expected ‘uint8_t *’ {aka ‘unsigned char *’} but argument is of type ‘const unsigned char *’
  566 | void rtl8168_rar_set(struct rtl8168_private *tp, uint8_t *addr);
      |                                                  ~~~~~~~~~^~~~
/var/lib/dkms/r8168/8.049.02/build/r8168_n.c: In function ‘rtl8168_init_board’:
/var/lib/dkms/r8168/8.049.02/build/r8168_n.c:25161:14: error: implicit declaration of function ‘pci_set_dma_mask’ [-Werror=implicit-function-declaration]
25161 |             !pci_set_dma_mask(pdev, DMA_BIT_MASK(64)) &&
      |              ^~~~~~~~~~~~~~~~
/var/lib/dkms/r8168/8.049.02/build/r8168_n.c:25162:14: error: implicit declaration of function ‘pci_set_consistent_dma_mask’ [-Werror=implicit-function-declaration]
25162 |             !pci_set_consistent_dma_mask(pdev, DMA_BIT_MASK(64))) {
      |              ^~~~~~~~~~~~~~~~~~~~~~~~~~~
/var/lib/dkms/r8168/8.049.02/build/r8168_n.c: In function ‘rtl8168_resume’:
/var/lib/dkms/r8168/8.049.02/build/r8168_n.c:28658:32: warning: passing argument 2 of ‘rtl8168_rar_set’ discards ‘const’ qualifier from pointer target type [-Wdiscarded-qualifiers]
28658 |         rtl8168_rar_set(tp, dev->dev_addr);
      |                             ~~~^~~~~~~~~~
/var/lib/dkms/r8168/8.049.02/build/r8168_n.c:24188:26: note: expected ‘uint8_t *’ {aka ‘unsigned char *’} but argument is of type ‘const unsigned char *’
24188 |                 uint8_t *addr)
      |                 ~~~~~~~~~^~~~
cc1: some warnings being treated as errors
make[1]: *** [scripts/Makefile.build:288: /var/lib/dkms/r8168/8.049.02/build/r8168_n.o] Error 1
make: *** [Makefile:1844: /var/lib/dkms/r8168/8.049.02/build] Error 2
make: Leaving directory '/usr/src/linux-headers-5.18.10-76051810-generic'