After seeing exactly what their concerns are, it seems that he needs more risk added to his kit, in addition to the late-game scaling and other things necessary to make him viable. He also probably suffers from some "victory through ignorance", in that he only does well when the opponent is ignorant of the long term danger that his turrets possess. All these problems need to be addressed. I could work out the math on what numbers would be required but instead I will just give generalities whenever necessary.
I would start by having his passive give mana rather than health and reducing the range to 350-400. This nerfs his early game sustain, indirectly nerfs his turrets (since they won't regen), but also makes his passive useful throughout the game. The reduction in range also adds some risk to his passive, since it forces him to clump up with his allies for long periods of time in order to gain useful effect and prevents him from remaining too far from the action.
I would cause his turrets to use mana per shot, which would require Heimer to remain close to his turrets or else have them run out of shots (15-20 seconds worth of shooting seems reasonable) thus adding another aspect of risk into his kit. With this change there would be much less difficulty with fixing the other problems by increasing the range of the turrets and letting them scale at a 1.5 ap-to-health ratio with an innitial HP of 150 (thus making them a little less defensive as a caster minion at lv 1 without AP but scaling better into late game) in replacement for the health they currently get per Heimer level. He could even have his third turret back (replacing the extra health level and moving the fire down to lv4) because using the turrets would be dangerous due to the necessity to be close and the gold given on destruction. The increased squishiness at low levels would also stop his low level dragon attempts but the changes would allow him to participate (but not solo) in an attempt at barron. See ulti for further changes, including a increased mana capacity per ulti level (allowing them to fire for 5 more seconds per level before needing refueling). The ulti would still heal the towers and give slowing shots. I would also change the firing particle to make it much more obvious what the turrett was attacking so that those unfamiliar with Heimer would understand the obvious counter to being attacked by the turrets. Everything else would remain the same.
I would cause his W to start out with 2 missiles, but adjust the damage to increase the low level base damage and raise the AP scaling (.05 should be enough). Each level of his ulti would permanently give him an extra missile (increasing to 2+3=5 at lv3) and activating his ulti would apply the same slow as is added to the turrets. The spell would also be correctly flagged as AoE rather than single target for item purposes.
I would reduce the particle speed of his E slightly but massively increase the scaling. I would also cause him to fire the missile at the beginning of the channel as opposed to the end, making him more able to hit at close range but not removing the penalty associated with the use. In addition his ulti would passively increase the maximum range and actively cause it to stun in the entire AoE, in addition to the normal effect.
I would first remove the chanel time on his ulti and increase the duration per level to 5-10-15 seconds (cooldown would need to be adjusted to compensate). I would then cause the ulti to have both a passive and an active effect on each of his other spells. Passively it would increase the mana pool on his turrets to the point where it could continue firing for 5 more seconds per level; it would increase the number of rockets by 1 per level; it would increase the maximum range of his grenade per level. Actively it would heal his turrets and add a slow to both his turrets and rockets, it would also cause the entire grenade area to stun on hit. I would also remove the passive CDR and just lower the cooldown on his abilities to compensate.
This increases the ability for Heimer to get in there and make big plays, allowing him to be much more aggressive. Every ability except his rockets would be changed to promote a much better risk/reward dynamic; passive encourages clumping and remaining in the center of the party, "Q" requires you to remain near the front of the battle to continue working, "E" will require you to be close to hit someone who is paying attention due to the slower speed but will be satisfying when it does. To a lessor degree the reduction of the low level missiles will also require better positioning in order to poke effectively. Since he will have a similar level of farm, due to his early game turret and passive nerfs, the hits should be much more on-par with other pokers and so should not be any more of a problem than theirs are.
In addition, besides the change that causes his turret to use mana and the change to the turret attack animation to make it more obvious, the rest of the changes are extremely straightforward in terms of programming and require no additional graphics work. Testing would likely take a month or so, but the rest could be put in place almost immediately.
These are just my current ideas and are likely to change. I'm still struggling with how to make his turrets long-term (ie: not temporary) but prevent them from being a drop-and-run type of high rewards/low risk ability; because as long as they are viewed as having that problem their late game scaling, squishiness, and range will almost certainly remain untouched. As long as he remains a super-farmer early game, his ranged abilities will always seem overpowered, so that is the other thing that needs to be changed.
Ideas and feedback are always welcome.
With that new turret change should they still get that short buff to attack speed at deployment? with possible free bullets during that time.