I’ll cover two of the possible policies that can be stable: interest rates and fiscal policy.
I will assume that a “stable” interest rate policy is to keep the policy rate (e.g. Fed Funds in the United States) unchanged. This is is a case discussed in “mainstream” economics, and the argument is that such a policy leads to inflation drifting at random. (As in Section 2.4.1 of “Monetary Policy, Inflation, and the Business Cycle” by Jordi Galí.) In other words, if you lose monetary policy, you lose control of inflation.
Of course, the policy rate has been at the same level for years in many countries, and not much happened. The reason why the policy rate was stable was usually that the economy was growing slowly, and no inflation risk was seen. That is, the economy was in a sense (temporarily) “stable”, and policy rate reflectd that.
Even if the central bank says it will not change short rates, bond yields would likely still move around on the basis that the policy will not be sustained. So things like mortgage rates would still move.
Some critics of “mainstream” economics think that the models that lead to this inflation instability are incorrect. In particular, some proponents of Modern Monetary Theory argue in favour of locking the policy rate at 0% forever. That is very much a minority position.
If we turn to fiscal, I am unaware of any formal theory. In basic “mainstream” models, fiscal policy is often left unchanged. However, this was not viewed as having much if an effect, since monetary policy is believed to be the main policy variable.
We have seen unchanged tax rates for years in many countries, and I do not believe anything unusual was associated with that. However, taxe rates are only part of fiscal policy. Spending changes from year-to-year, and in the real world, that is unlikely to change. For example, the Canadian Federal Government is helping pay for a bridge near where I live. That spending will hopefully cease when that construction is complete.
To top it off, the dollar amounts of deficits vary based on the business cycle. If people lose jobs, they pay less taxes and end up receiving more governmental assistance. So if someone defines fiscal policy “stability” as an unchanged fiscal deficit, that is almost guaranteed to never happen. However, these changes to the deficit are believed by some economists to help dampen the business cycle (the “automatic stabilisers”). (I see little discussion of automatic stabilisers in modern “mainstream” theory, but it was discussed more by the old Keynesians.)
In summary, “stable” policy settings imply that the government cannot lean against fluctuations like recessions, or control inflation. Economists of the Keynesian persuasion would likely argue that this will result in deep depressions, such as in the 1800s. Many non-Keynesians would dispute that.