Reduced immigration may ease rental demand in some places, but the evidence does not show that deportations are causing significant nationwide declines in both rents and home prices. Other forces, especially new apartment supply, are also driving rent changes, while reduced construction labor could push home prices higher.
Deporting people out, what does that do? That lessens pressure on the housing price and rental market, and so we're seeing significant drops in both housing prices and rents.

Speech by Governor Kugler on a view of the housing market and U.S. economic outlook - Federal Reserve Board
Source: federalreserve.gov
Key Evidence
Federal Reserve research indicates that immigration can raise rents in the short run, but it does not establish a matching, policy-specific effect from deportations.
Newsweek reported that housing demand was easing after fewer immigrants arrived, but this is not evidence that deportations alone caused significant declines in both rents and home prices.
Reporting in The Washington Times attributed falling rents to several factors, including a large number of newly built apartments, while Investopedia noted that losing construction workers could reduce housing supply and put upward pressure on home prices.
What the Evidence Shows
The evidence supports only a narrower point: immigration can add short-term pressure to rental demand. Federal Reserve Governor Adriana Kugler said research finds that rent growth can increase after an immigration wave, although that effect may fade as housing supply expands. That research does not establish that removing immigrants will produce significant declines in rents and home prices, nor does it measure the effect of a deportation campaign.federalreserve.govfederalreserve.govSpeech by Governor Kugler on a view of the housing market and U.S. economic...
federalreserve.govfederalreserve.govSpeech by Governor Kugler on a view of the housing market and