What does the shift to dual lanes mean for the 1-5 farm priority system?

Nothing changes really. In pubs you often play by ear who is designated 4 or 5 based on the hero impact and item requirements. ( a leshrac is more item dependent than a rubick) etc.

However teams have this stuff already lined up and decided before the match begins. The gold-priority system of 1-5 isn't affected by lane setups. You still need 1 hero to specialise in consumable support items such as wards sents dusts, so that the other support can get items utility items. So having your 4 in the safelane near jungle pulls/stacking for gold and potentially easier ganks is still the way to go.

*just to clarify, the 1-5 system is to allow specialisation which is much more gold-efficient and time-efficient than with just sharing the costs evenly. Imagine for example that 2 supports share costs equally, and by 20 minutes both supports do not have crucial utility items up (force, mek etc) and as a result any teamfight taken is much more difficult or completely lost. So both heroes are screwed and have no impact.

Now imagine if you specialise the roles and by 15 - 20 mins those supports have a force + mek or urn or a really early mek gets completed and turns every teamfight or gank into a sure win. now you can push and punish easily right?

The 1, 2, 3, 4, 5 possition terms lost their strict meaning some time ago. These days they're commonly used to simply describe possitions like other sports. As an example the solo mid is always referred to as 2 even when the hero is often 3rd in farm priority. A further example is stuff like the WPC-ACE player profiles which define 1 as carry 2 as solo mid 3 as solo offlane 4 as jungle/support and 5 as hard support.

I feel that offlane is getting a lot more farm, shifting that person closer to 2nd position, instead of 3rd and 4th being vaguely close to eachother because of how easy it was to shut down a single hero with a trilane in previous patch.

All in all, the new patch gives everyone more gold because of the increase in the passive gold given over time. The Midas-spree definitely helps as well, allowing poor supports to get even better items, items which they before would probably never get (AA Aghanims, CM Blink/Bkb, Disruptor Aghanims).

The positions have always been kind of weird and are always able to shift around. A lot of people in the pro scene (casters and players alike) have mentioned that position and farming priority numbers aren't something that is really important and objective.

Also to try and answer your question I guess the most obvious part is that mid will mostly be the 2 position in a dual lane matchup. The rest depends on the other team's lanes and how agressive your lane is. Most of the times you can already know from the drafts which lanes are going to win and which are going to lose, that is how you determine your farm priority.

1-5 is farm priority, not position. Farm priority will remain the same, but the way teams earn that may differ. In a dual lane situation, it looks more like:

1. Safe lane farmer
2. Mid
3. Long lane farmer
4. Long lane support
5. Safe lane support

Obviously this is variable, but it is one example. The point is just that position is somewhat independent of farm priority-- they're correlated, but shift with the metagame. For example, some mid heroes consume more farm than others, which can move mid up and down the priority. Recently the system has been somewhat abused, but this will shake it up a bit.

Farm priority stays the same. The big difference is the offlaner gets a lot more from the laning phase now, instead of being level 1 at 4 minutes and playing catch up. As Aui mentioned, this frees up room for the supports to get extra farm because teams don't have to designate so much space to letting the offlane catch up

Nothing.

The old system actually had a problem in that the 4 (pulling/jungling support) often got more than the 3 (offlaner). Now the offlaner gets plenty, but the 4 gets less due to the pull camp changes. The gap in farm between the 3 cores and support duo is larger, but the increase in passive gold means supports aren't completely starved. So I'd say that in most set-ups the solo mid is still 2, offlane farmer 3, jungler/safelane pull support is 4, and the other support (who can be in any of the three lanes) is the 5.

Because of the off-lane changes, you can get 3 farmers in every lane and a possible jungler. Before 6.79, off-lane farming was difficult/risky and the off-laner posed an increased risk of dying because of over-extension.

Now that the side lane creeps meet closer to the off-lane tower, the off-lane has more potential to be a farming lane rather than just an aggressive tri/defensive solo lane.

Laning number doesn't have to equal farm priority. Unless you have a mid ta, sf, invoker, od and etc often your team will have an offlane/secondary carry core that benefits from farm more than your mid. If you do go a more carry mid teams will then more likely pick up more utility offlaner like clock, et, wr and bounty. Many mids are picked up to get solid levels and a key item or two and then to create space at which point their own farming level may be less important.

Nothing?

4 takes as much farm as 5, 3 takes that surplus