Group by maximum consecutive row
In Postgres (tested with v9.3) you can use the dedicated inet data type, to store IPv4 addresses with only 7 bytes (or IPv6 with 19 bytes) and with automatic integrity checks and dedicated functions and type casts etc.
Schema
The translated schema could look like this:
CREATE TABLE log (
  id serial PRIMARY KEY
, dst_port int
, src_ip inet
, dst_ip inet
);
CREATE INDEX ON log (dst_port);
CREATE INDEX ON log (src_ip);

I moved to dst_port int to the 2nd position to optimize alignment / padding:

Configuring PostgreSQL for read performance

Now we can use standard window functions (not possible in MySQL).
Step 1: Fold groups of consecutive dst_ip for same (dst_port)
One special difficulty: The aggregate function min() / max() are not yet implemented for inet in Postgres 9.4. Both are in the upcoming Postgres 9.5!
So I substituted with DISTINCT ON in the first step:

Select first row in each GROUP BY group?


SELECT DISTINCT ON (dst_port, ip_grp)
       dst_ip, count(*) OVER (PARTITION BY dst_port, ip_grp) AS ip_ct, dst_port
FROM  (
   SELECT dst_ip, dst_port, dst_ip - row_number() OVER (PARTITION BY dst_port
                                                        ORDER BY dst_ip) AS ip_grp
   FROM   log
   ORDER  BY dst_port, dst_ip
   ) sub
ORDER  BY dst_port, ip_grp, dst_ip;

Result as desired - with a count of rows (could be upper IP as well).
You can subtract/add integer from/to the inet type. By subtracting the row_number() all consecutive rows get the same grp - the value of grp is irrelevant, just the fast that it's the same per partition (dst_port).
Then we can GROUP BY ... - or in this special case DISTINCT ON dst_port, ip_grp. I use another window function to get the count ip_ct in the same step: count(*) OVER (PARTITION BY dst_port, ip_grp) AS ip_ct.
Note that consecutive IPs can cross byte boundaries (see my comment to question).
Detailed explanation for this technique:

Select longest continuous sequence

Step 2: Fold groups of consecutive dst_port for same (dst_ip, ip_ct)
SELECT dst_ip, ip_ct, min(dst_port) AS dst_port, count(*) AS port_ct
FROM  (
   SELECT *, dst_port - row_number() OVER (PARTITION BY dst_ip, ip_ct
                                           ORDER BY dst_port) AS port_grp
   FROM  (
      SELECT DISTINCT ON (dst_port, ip_grp)
             dst_ip, count(*) OVER (PARTITION BY dst_port, ip_grp) AS ip_ct, dst_port
      FROM  (
         SELECT dst_ip, dst_port, dst_ip - row_number() OVER (PARTITION BY dst_port
                                                              ORDER BY dst_ip) AS ip_grp
         FROM   log
         ORDER  BY dst_port, dst_ip
         ) sub1
      ORDER  BY dst_port, ip_grp, dst_ip
      ) sub2
   ) sub3
GROUP  BY 1, 2, port_grp
ORDER  BY 1, 3, 2;

Basically, repeat the same logic like in the first step, applied to the result of the first step.
But now you have to group on ip_ct additionally. And this time, you can use the simpler min(dst_port), since the port number is a plain integer.
SQL Fiddle demonstrating all.