It may help to consider the diagram below which shows the percentage saturation of haemoglobin with oxygen (and hence how much it takes up or releases) at the different partial pressures in different tissues. (The actual pattern — and values — depends, of course, on the blood flow to different tissues.)
In general, the curves show that at the high oxygen pressure of the lungs haemoglobin is near saturated with oxygen — Hb takes O2 up —, whereas at the lower oxygen pressure in the peripheral tissues the saturation is less — Hb releases some O2. So in normal circumstances oxygen will be gradually released to tissues as it circulates through the body. The oxygen pressure in the tissues is acting as the control valve for release.
In cases where there is a vastly increased demand for oxygen, this normal control is insufficient. The hydrogen ions in the acids produced by respiration (carbon dioxide and perhaps lactate) act as an additional control as they cause the oxygen dissociation curve of haemoglobin to change e.g. from 1 to 2 in the diagram. (The molecular mechanism of this is understood, but that is not the question.) Hence at the same oxygen pressure, more oxygen is released (cf. the percentage saturation at 1 and 2).
Does this harm tissues other than the ones producing the acid? They obviously have lower priority, but you can see there is still oxygen left at 2, and physiological mechanisms (faster heart beat etc.) will increase the blood flow in general. So it is wrong to think of this as ‘all or nothing’, but as part as a general response which alters the balance towards the tissues in greatest need.