What was the status of Shia Muslims in the Ottoman empire?

This is very complicated for several reasons: (1) the time period is long, (2) regional / local factors and (to varying degrees) perceived external and internal threats meant changing policies, and (3) tolerance did not usually mean equality. Perhaps this is at least one reason for the following scholarly observation:

While a tremendous amount of scholarly material is available on the history of the Ottomans, surprisingly little of a general nature has been written on the history of Islam in the Ottoman Empire.

The Encyclopedia of the Ottoman Empire has a summary of Shia (or Shii) Islam during Ottoman rule:

Before the 16th century, the Ottomans were relatively uninterested in the differences between Shii and Sunni Islam. The Sufi orders (see Sufism), which put Ali and the prophet’s descendants at the center of their devotions, were much more popular in the early Ottoman Empire.

Source: Gabor Agoston & Bruce Alan Masters, 'Encyclopedia of the Ottoman Empire'

However, this did mean not equality, and the Ottomans were concerned with the spread of Shia Islam:

Shiʿism and so-called Islamic heresies were major internal issues as well as an external threat for the Sunni Ottomans. One means of curbing Shiʿism, as well as promoting Sunni Islam, was through the patronage of the judicial system that was organized and formalized in a new manner by the Ottomans.

Also, the Ottomans were not prepared to just sit back if those adhering to Sunni Islam came under attack:

...in 1501 Shah Ismail I (r. 1501–24) made Imami Shiism the religion of his court in Iran and began to persecute Sunni Muslims in his realm. In addition, the Turkoman tribes of eastern Anatolia rose in rebellion against the Ottomans, proclaiming Shah Ismail to be the long-awaited imam who would restore justice to the world. These rebels were known as the Kizilbaş in Ottoman texts and were ruled as heretics by the Ottoman Sunni legal establishment, making it legally permissible for the Ottoman army to kill them. The Shii rebellion against the Ottomans was temporarily ended by the Battle of Çaldiran in 1512.

Source: Agoston & Masters

Thus, due to the presence of hostile Shias to the east and

and the ongoing anti-Muslim challenges from the Christian world, the Ottoman sultans were increasingly motivated to present themselves as the upholders and defenders of what they regarded as the true faith of Sunni Islam.

Source: Agoston & Masters

Despite officially moving more firmly towards Sunni Islam, persecutions of adherents to Shia Islam did not generally happen to those

who accepted Ottoman rule peacefully, even if they held the view that such rule was illicit in the absence of the imam. This tolerance was especially manifested in the Ottoman treatment of the Shia of Iraq. When Süleyman I (r. 1520–66) conquered Baghdad in 1535, for instance, he endowed Shii shrines as well as Sunni ones and hosted Shii clergy along with their Sunni counterparts. Subsequent Ottoman governors extended this strategy of tolerance and even provided patronage to Shii shrines and clergy. In part, this was due to a realization that it was best not to alienate the Shii subjects of the sultan in Iraq. But it also seems to have arisen out of a more general cultural practice of tolerance that the sultans extended to both non-Sunnis and non-Muslims.

Source: Agoston & Masters

Nonetheless, Shia muslims within Anatolia did undergo a period of repression during the 16th century on the basis of heresy. This repression peaked in 1570s:

Normally, official charges of Qizilbash heresy contained in the Mühimme sources ranged from a refusal to attend the mosque or follow the shan’ a, to insulting the Sunnis and the ’ulema, playing music in the tekkes (convents), reading or possessing “heretical” books, taking part in orgies, collecting alms for the Safavids and secretly working for them as khalifas.

Source: Fariba Zarinebaf-Shahr, 'Qızılbash “Heresy” and Rebellion in Ottoman Anatolia During the Sixteenth Century'