Support for DKIM signing in Exim is available since version 4.70, and the configuration supplied with Debian makes it fairly straightforward to implement. However it suggests an all or nothing configuration wherein all outgoing mail is signed with the same domain authority.

Where multiple domains are used it may be necessary to selectively switch on DKIM signing, and be able to specify the signing domain. The following details provide a mechanism to do so within the standard Debian Exim configuration.

(This assumes that the keys have been created and the requisite records have been added to DNS for the affected domains. It also assumes a split config.)

Set up a simple look up file such as /etc/exim4/dkim_senders

*@example.com: example.com
test@example.org: example.org

This config should mean that anything sent from any address at example.com is signed as example.com, but only test@example.org will be signed with the example.org key. If default DKIM is not enabled, then no other example.org mail will be signed.

Now create a new router that sits in front of the main router for external main (whatever uses remote_smtp as a transport e.g. dnslookup) such as /etc/exim4/conf.d/router/180_local_primary_dkim (basically a copy of dnslookp with a modified transport)

dnslookup_dkim:
  debug_print = "R: dnslookup_dkim for $local_part@$domain"
  driver = dnslookup
  domains = ! +local_domains
  senders = lsearch*@;/etc/exim4/dkim_senders
  transport = remote_smtp_dkim
  same_domain_copy_routing = yes
  # ignore private rfc1918 and APIPA addresses
  ignore_target_hosts = 0.0.0.0 : 127.0.0.0/8 : 192.168.0.0/16 :\
                        172.16.0.0/12 : 10.0.0.0/8 : 169.254.0.0/16 :\
                        255.255.255.255
  no_more

Then add in a new transport /etc/exim4/conf.d/transport/30_local_remote_smtp_dkim (basically a modified version of remote_smtp)

remote_smtp_dkim:
  debug_print = "T: remote_smtp_dkim for $local_part@$domain"
  driver = smtp
.ifdef REMOTE_SMTP_HOSTS_AVOID_TLS
  hosts_avoid_tls = REMOTE_SMTP_HOSTS_AVOID_TLS
.endif
.ifdef REMOTE_SMTP_HEADERS_REWRITE
  headers_rewrite = REMOTE_SMTP_HEADERS_REWRITE
.endif
.ifdef REMOTE_SMTP_RETURN_PATH
  return_path = REMOTE_SMTP_RETURN_PATH
.endif
.ifdef REMOTE_SMTP_HELO_DATA
  helo_data=REMOTE_SMTP_HELO_DATA
.endif
dkim_domain = ${lookup{$sender_address}lsearch*@{/etc/exim4/dkim_senders}}
dkim_selector = yourhostname
dkim_private_key = /etc/ssl/private/dkim.key
dkim_canon = relaxed
dkim_strict = false
#dkim_sign_headers = DKIM_SIGN_HEADERS

I've left the selector and keys the same since there doesn't appear to be any problem sharing these across domains, but these could also be found via lookups if needed.