Outline:
Although we wouldn't be around long enough to see it, the extinction of heterotrophic bacteria would ultimately be the boon of terrestrial fungi and marine wood borers.
Upon extinction of the heterotrophic bacteria, marine phytoplankton in all size classes would see a dramatic decrease in competion for dissolved organic matter (DOM) and dissolved inorganic nitrogen (DIN). Since most of the oceans are nitrogen limited, the abundant dissolved resources might induce a global bloom, followed by a zooplanktonic feeding frenzy. In only a few days, the planet would flash a green alarm.
Simultaneously, protists and animals on land would be having a rough time. Their food source, the primary producers of terrestrial ecosystems, would quickly suffer from the lost decompositional action of the heterotrophic bacteria. Fungi could potentially take up some slack, but the standing stock of macrophytes would likely fall precipitously. For those animals and protists that depend on heterotrophic bacterial symbionts to assimilate nitrogen for vital protein synthesis (Jumars, 1997) , serious stomach aches would be the norm, and death would be imminent. A similar fate would befall all marine animals within weeks. Interestingly, humans might escape by degrading their food sources through thorough thermal processing and gleaning essential amino acids and vitamins through industrial production. While all the dead bodies would decompose to some extent (through solar irradiation, physical degradation via erosion and wave action, for instance), a huge quantity of organic material would be transported to the sea floor. Marine animals would become sedimentary deposits within weeks (starvation), and terrestrial animals would be buried on land within weeks, or transported to the sea floor over centuries . No remineralization by microbes would take place as the particles or superstructures sank through the water column; any remineralization by fungi would have to take place on land; there would likely be a lot of woody debris in the oceans, potentially available to the wood-boring marine species.
Also in the first thousand years, a huge amount of organic material -- a million dead bacterial cells per milliliter of seawater -- would await decomposition or consumption. Populations of bacteriovores might subsist solely near upwelling regions, limited exclusively by their encounter rate with their (exponentially decreasing, extinct) prey. Their demise might be much accelerated because of their inability to make net energetic profits off concentrations less than about 10,000 - 100,000 cells per milliliter (Jumars, 1993). Additionally, they would compete with any process which destroyed bacterial cells: exposure of membranes to ultraviolet radiation; subjection to intense heating within hydrothermal systems; or, osmotic transport of organic contents. Since viral infection of dead cells is impossible, the remnants of the extinct bacteria might persist for a suprisingly long time (centuries?). If solar degradation was effective in the surface layer of the ocean, however, all traces of both bacteriovores and their prey would be erased after about a millenium (residence time of oceanic circulation).
As the initial bloom subsided due to a rebalancing of phytoplanktonic growth and grazing, a change in the composition of the dissolved organic matter would occur. Chemoautotrophic bacteria would continue to oxidize ammonium, but denitrification would cease. A fraction of the nitrogenous wastes and dissolved organic matter excreted by zooplanktonic predators (and predators at higher trophic levels) would be directly available for use by phyoplankters, but essential inorganic nutrients and carbon bound to particles or aggregates would begin to fall into the deep ocean. Sinking particles would reach the sea floor with minimal bacterial degradation, as they do today (Suess, 1988). Over the course of years, the nitrogen compounds circulating in the oceans' water -- crucial compounds for all osmotrophs, and most importantly photosynthetic ones -- would shift away from molecular nitrogen, and wane in concentration due to sequesteration in the sediments.
The crucial difference would occur at the seafloor. With heterotrophic bacteria, rapid turn-over of phytodetritus at the sea floor likely controls the oxygen and nutrient chemistry of the oceanŐs interior (Suess, 1988). Unless chemoautotrophic archaea or bacteria could remineralize the raining organics as effectively as before extinction, the surface ecosystems of a planet without heterotrophic bacteria would see a long-term decrease in nitrogen and carbon stocks. Subduction would ultimately return some of the material to the top (in terms of elevation -- volcanoes!) of the terrestrial ecosystems, but the fluxes and stocks would be much reduced. After millions of years, at least an entire kindom of life would be gone, the trophic structures of all ecosystems greatly simplified, and the oceanic availability of organic building blocks greatly reduced.
References:
Jumars, P.A. 1997. Class notes -- marine nitrogen cycle handout.