A litigant could issue a subpoena to you demanding the information on your phone relevant to their case.
If your employer or you are parties to a lawsuit, you can also be required to provide information through what is a called a "request to produce" issued by one party to another party without a subpoena, and under general information disclosure requirements in some lawsuits that apply to parties and people affiliated with them. For purposes of this answer, a subpoena, a request to produce, and a disclosure obligation of some other kind are all basically equivalent in the ways that matter.
The search warrant process in criminal investigations, however, is quite different, but this is rarely used in the contexts that you are concerned about in your question.
Usually, the obligation would be to provide a copy of the relevant data (e.g. emails and text messages and phone call history and metadata related to the case), rather than to turn over the phone itself, and this would usually involve a couple of weeks notice or more.
Absent extreme circumstances (e.g. you are a flight risk defendant, or only technical means can get access to secret malware not accessible to an ordinary user on it), you are usually allowed to participate in sorting out what is and is not responsive to the request and they cannot just demand "everything on your phone."
If the subpoena demand is overbroad, you (or perhaps your employer's attorney), can move to quash the subpoena or limit its scope before you have to comply in the court that issued the subpoena. You could also make objections based on, for example, attorney-client privilege or the 5th Amendment right against self-incrimination, in an effort to object to the subpoena's demands for information.
Absent a criminal case where you are a suspect, it would be very rare to actually seize the entire phone itself or to search all of its contents without your involvement.
Note that this is a risk that almost everyone who communicates about business by phone faces.
For that matter, even people who don't communicate about business by phone face this risk.
For example, suppose you see a plane crash into a house while you are on the phone with your brother talking about last night's baseball game, and then mention that fact when interviewed by police investigating the crash that you witnessed. You and your brother's metadata from your calls with each other from that afternoon could be subpoenaed, in connection with a wrongful death or property damage lawsuit arising out of the crash, to help determine precisely when the plane crashed.
It doesn't matter who owns the phone. Usually the subpoena will apply to all accounts and devices within your possession, custody and control. So, even if you don't have the phone in your possession when you are served with the subpoena (e.g. your son borrowed it for the week), and even if you made a call on your spouse's phone, if you have the legal right or practical ability to get the information, you must provide it.
The apps installed on the phone don't matter. What matters is whether there is information relevant to the case (or potentially relevant to the case) that can be accessed with your phone, and whether the subpoena has asked for it.