I think this is a multifaceted question, that needs an answer for each animal or plant type. So I'll consider humans as the main case then breify a few other examples.
If the increase was substantial, say greater than 2g, I doubt that any humans would survive for long. A few hardly individuals might last a few weeks or even a few months but the casualty rate would be terrible. Older people often break bones when they fall, in part because they can’t react quickly enough and in part because their bones are weaker than those of younger people. In a 2g environment even the young would suffer these problems because their reaction times would have to be so much faster to prevent damage (by extending arms etc) and the increased speed of impact would make their bones more likely to break.
At 1.5g it’s a tougher call, but I suspect not in part because of the reasons given above for 2g conditions, but also because of the break down in society caused by multiple dislocations to civilization. Huge numbers of people in hospital due to fractures and many other complaints exacerbated by the increased gravity eg asthma. Any mechanism that relied on gravity in any way might change its behaviour or stop functioning. Such things might be hard to predict but would range from the inconvenient to the seriously damaging for example doors becoming jammed or hard to open, most, if not all, planes grounded and many plants like modern wheat ending up bent over and hard to harvest.
So at 1.5g I would say it’s borderline but non-survivable due to general collapse. Much less than that I think it’s survivable by at least some for years.
At 1.2g I suspect a lot would survive for long enough to ensure the continuity of the species. The young would have the advantage (if that’s the right word) of growing their bodies in the new gravity so would almost certainly be better adapted than their parents even without any genetic adaption. Beyond that natural selection would ensure that those best capable of adapting would survive in the greatest numbers and the human genome would be whittled down a fair bit (less tall people more short people etc) leaving just the fittest.
Over the longer term evolutionary adaption by mutation would become an important factor but might take hundreds of thousands of years.
Birds are much more vulnerable to gravitational issues and would be seriously affected by any change more than a few percent with the heaviest birds suffering most. It is very hard to say exactly but I would estimate somewhere between 1.05 and 1.1g would see off all flying bird species.
Fish should be less affected due to buoyancy, but might still struggle with swim bladder issues. 1.5-2g?
Large species or those with fine structures would be seriously affected and would be first to go the more sturdy squat varieties would last longer and ground hugging species would survive longest.
Things like algae, lichen, plankton and bacteria would all be affected but should be capable of withstanding much higher g forces. It is hard to predict but I suspect 5-10g would kill many even of these species. The ultimate survival limit is almost impossible to say as the world would be a very different place.
Evidence and references
There is some information of interest in these references
But I doubt that there is any really hard evidence available because the situation is inaccessible to any form or simple experiment. So this is a matter of opinion to some degree. Although the g forces quoted in the second reference might appear to suggest that 4g is doable, bear in mind this is for astronauts in acceleration couches. Not for joe public trying to do his shopping and climb stairs etc.This reference tends to support my view about injury (page 53). I would suggest this is especially true when not lying on an acceleration couch.
“During dynamic flight phases, there is potential for impact and flail injury, which includes crewmember extremities impacting vehicular surfaces or objects, hyperextending, hyperflexing, hyper-rotating, fracturing, or dislocating if proper restraints and supports are not used. Features such as harnesses, form-fitting seats, and tethers may help maintain the proper position of the crewmember's body and limbs to reduce movement or contact with vehicle surfaces that would produce injury. In addition, the design of spacesuits may contribute to reducing injury to the crew. Preventing the inadvertent contact of extremities with vehicular structure or interior components significantly reduces the likelihood of limb fracture or soft tissue injury during a dynamic flight event. Extremity guards, tethers, garters, and hand holds have been used to reduce injury in other spacecraft, aircraft, and automotive vehicles”